To my understanding, when the King James Bible was translated, it was translated into how English was spoken at the time. However, the Book of Mormon, having been allegedly translated in the early 1800s, was not translated into how English was spoken in the 1820s but rather into a form of English very similar to that of the King James Bible. Is there any cited reason - whether spiritual or not - for this apparent stylistic choice?
This is a great question and to answer it it's important to understand that for most of its history discussions about the LDS Church have been divided into two neatly bifurcated camps: members who insist that the church is true, by which they mean that Joseph Smith's account of how he came to translate the Book of Mormon and found the church is real and accurate, and critics of the LDS Church who insist, often with great exasperation, that Smith's claims are obviously nonsense. The Book of Mormon has often been at the heart of this division.
The short answer to your question is no, there was no cited reason at the time. There wouldn't be because when the book first appeared, those who believed in it saw a book of scripture that sounded like the Bible. To them, it made sense. But since then critics have used the presense of King James English to argue that the book is not authentic, and was in fact invented and written by Joseph Smith. LDS apologists over the years have responded with various defenses of why the book might include King James English. Here's some background:
According to Joseph Smith, the Book of Mormon was translated from gold plates given to him by an angel in upstate New York. The Book of Mormon, published in 1830, tells the story of a family who left Jerusalem in 600 BCE, built a boat, and sailed to the new world. From there the family split into two warring factions, the Nephites and the Lamanites, named after two brothers. The book tells of their history for the next ~1000 years. There are wars and periods of righteousness and wickedness among both groups, which grow into massive civilizations. Throughout the book, there are references to cities, animals, monetary units, and geography. After Jesus' death and resurrection in the New Testament, there is an account in the Book of Mormon of Jesus visiting the Nephites in the New World. He invites them to feel the wounds in his hands and in his feet. He preaches to them. The Book of Mormon is a holy book to Latter-day Saints, and this is perhaps the most important part of the scripture. By the end of the book, a major war between hundreds of thousands of people has broken out that has largely wiped out the Nephites, leaving only the Lamanites. A prophet in the book, Mormon, leaves the record of the Nephites on gold plates to his son, Moroni. Moroni buries the plates in a hill he calls Cumorah. Smith would dig these plates up from the Hill Cumorah in Manchester, New York, in 1827, and translate them into the Book of Mormon. As we will see, even the word "translate" has become a sticking point.
From the beginning, the history the Book of Mormon purported to tell drew criticism and scorn. I won't go into all of the issues here; suffice it to say that assertions that the Book of Mormon is a made up book by Joseph Smith that can't possibly be a real history is perhaps the biggest criticism leveled against the LDS Church by other Christian religions (and some academics) that take issue with LDS beliefs and theology. There are criticisms of geography, criticisms of population sizes, criticisms of the animals mentioned in the book, etc. There are, quite literally, hundreds of books on these topics. But there have always been LDS apologists and defenders of the church who respond to these criticisms, and some of them involve your question on the kind of English used in the Book of Mormon.
Until recently, the LDS Church has maintained that Smith's translation of the gold plates was a literal translation—that the words found in the Book of Mormon are the same as those that were found on the gold plates, albeit in what Smith said was "Reformed Egyptian." And throughout most of its history, critics have challenged that assertion in various ways. One of the main criticisms is that Smith plagiarized the King James Bible. There are, for example, lengthy excerpts from the Old Testament book of Isaiah in the Book of Mormon. Why, the question goes, is this in King James English? And for that matter, why is the rest of the book in King James English?
For critics, the answer is obvious: Because Joseph Smith wrote the book in an attempt to make it sound like the scripture he was familiar with, the King James Bible. Members of the LDS Church have responded to this with different ideas of varying orthodoxy. The most common is that God inspired Joseph Smith to translate the Book of Mormon into King James English because it was what readers in 1830 would have been familiar with. It is, they argue, the spiritual nature of the book that matters, not an exact translation. And in recent years the church itself seems to have more openly embraced this idea, since it has acknowledged that the way Smith translated the book was by placing a seer stone in a hat and putting his face into the hat to shield the light. He was, in other words, not translating so much as he was receiving revelation of what the book says.
Other members, although there is less acceptance of this, suggest that perhaps the history of the Book of Mormon isn't real at all. It's inspired fiction, meant to teach spiritual truths. Other people find a blend of these two theories, saying that the broader history is real, but the details don't need to be—again, it's a spiritual book meant to bring people to Christ.
These are a fraction of the issues your question raises. But to reiterate: critics have frequently used the presence of King James English to argue that Joseph Smith wrote the Book of Mormon; LDS apologists over the years have mustered various defenses of its presence in the book.
I'd recommend starting with Terryl Givens, "By the Hand of Mormon: The American Scripture that Launched a New World Religion" (New York: Oxford University Press, 2002). If you want to dive deeper, there are countless books and articles I can recommend. One that gets into the main criticisms of the Book of Mormon, from the perspective of a former Latter-day Saint, is Earl M. Wunderli, "An Imperfect Book: What the Book of Mormon Tells Us about Itself" (Salt Lake City: Signature Books, 2013). And an excellent resource for how the Book of Mormon came about is Larry E. Morris, "A Documentary History of the Book of Mormon" (New York: Oxford University Press, 2019).
Is there any cited reason - whether spiritual or not - for this apparent stylistic choice?
I am not aware of any official LDS Church response to this question per se, but responses by LDS scholars to this question have been published in Church owned venues/media in the relatively recent past.
The first of these was LDS scholar Hugh Nibley responding to a letter to the editor of The Church News published July 29, 1961. Some of that response has been produced on the FAIR website here:
... Now as to [the] question, "Why did Joseph Smith, a nineteenth century American farm boy, translate the Book of Mormon into seventeenth century King James English instead of into contemporary language?"
The first thing to note is that the "contemporary language" of the country-people of New England 130 years ago was not so far from King James English. Even the New England writers of later generations, like Webster, Melville, and Emerson, lapse into its stately periods and "thees and thous" in their loftier passages.
Furthermore, the Book of Mormon is full of scripture, and for the world of Joseph Smith's day, the King James Version was the Scripture, as we have noted; large sections of the Book of Mormon, therefore, had to be in the language of the King James Version—and what of the rest of it? That is scripture, too.
One can think of lots of arguments for using King James English in the Book of Mormon, but the clearest comes out of very recent experience. In the past decade, as you know, certain ancient nonbiblical texts, discovered near the Dead Sea, have been translated by modern, up-to-date American readers. I open at random a contemporary Protestant scholar's modern translation of the Dead Sea Scrolls, and what do I read? "For thine is the battle, and by the strength of thy hand their corpses were scattered without burial. Goliath the Hittite, a mighty man of valor, thou didst deliver into the hand of thy servant David."[3]
Obviously the man who wrote this knew the Bible, and we must not forget that ancient scribes were consciously archaic in their writing, so that most of the scriptures were probably in old-fashioned language the day they were written down. To efface that solemn antique style by the latest up-to-date usage is to translate falsely.
At any rate, Professor Burrows, in 1955 (not 1835!), falls naturally and without apology into the language of the King James Bible. Or take a modern Jewish scholar who purposely avoids archaisms in his translation of the Scrolls for modern American readers: "All things are inscribed before Thee in a recording script, for every moment of time, for the infinite cycles of years, in their several appointed times. No single thing is hidden, naught missing from Thy presence."[4] Professor Gaster, too, falls under the spell of our religious idiom. [A more recent example of the same phenomenon in the twenty-first century is discussed here.]
By frankly using that idiom, the Book of Mormon avoids the necessity of having to be redone into "modern English" every thirty or forty years. If the plates were being translated for the first time today, it would still be King James English!"
Brigham Young University's Religious Studies Center hosts the publication The King James Bible and the Restoration from 2011 which includes a chapter written by Daniel L. Belnap where he addresses this question:
Of the two questions, the first [Why does the Book of Mormon incorporate King James English?] is easier to answer, at least from a cultural perspective: the presence of King James English, or early modern English, in the Book of Mormon is most likely the result of the KJV's influence on religious discourse in the early nineteenth century. Such language, though no longer the spoken English, was used in sermons and discourses to impose a sense of tradition to what was being spoken and written. [3] In fact, as late as the early twentieth century, King James language was still used in certain translation mediums. For instance, Charles's translation of the Apocrypha and the Pseudepigrapha, Loeb's series of translations for ancient Latin and Greek texts, and other translations of Classical and Near Eastern texts were written in the King James English to instill an air of antiquity to the nonbiblical texts. That such language would be utilized for the Book of Mormon, a text that purported to be not only ancient but directly related to the Bible, is not a surprise.
... We do know that within the context of Joseph Smith's time, the use of King James English for religious purposes is neither unique nor strange but was common when one sought to establish tradition and authority for either the written or the spoken word. It does seem clear that the presence of King James English may have conveyed a sense of authority early in Church history because the early revelations recorded in the Doctrine and Covenants that were given before the Book of Mormon was completed, as well as Joseph's earliest account of the First Vision, appear also to have utilized King James English. [11]
The familiarity of Joseph's contemporaries with King James English would have made their reading of the Book of Mormon an easier experience and one in which they could more readily accept the claims made by the book as both scripture and as an ancient record that was tied to biblical history. The use of King James English in the Book of Mormon would itself be part of the Lord's promise that he would reveal his truth to humankind "after the manner of their language, that they might come to understanding" (D&C 1:24). The specific phrase "manner of language" suggests that such transmission would occur not only in terms of the actual language but even in specific mannerisms that would be recognized by the audience in question. That this "manner of language" worked, at least in part, can be evidenced by how quickly the Book of Mormon became known as the "Golden Bible" or "Mormon Bible" by believers and critics alike, demonstrating a tacit understanding that, at the very least, the Book of Mormon sounded biblical. Thus the presence of King James English and even KJV passages in the Book of Mormon functioned to establish the book's validity to people already familiar with the words of God via King James English while making it easier to recognize the truths found therein because of the text's familiar cadence and sound.
Outside of those two responses which were disbursed at one time or another via LDS controlled organizations, probably the most prominent response from an LDS scholar on this topic took the form of two blog posts written by Stephen O. Smoot and posted on The Interpreter Foundation website:
In particular, Smoot highlights a statement by David Webbington:
By the 1820s, the language of the early seventeenth century had become so much in vogue that the most popular preacher in London, Edward Irving, deliberately adopted its archaic idiom. “The whole Philosophy of Europe serveth infidelity,” as Irving once declared, sounded more powerful than if he had used the current form of the verb, “serves.” The characteristic of the times was not to modernize the English Bible but to imitate its accepted translation.
Notice the date. Joseph Smith was translating the Book of Mormon (while trying to “imitate” the “accepted translation” of the KJV, no less) at exactly the same time.