I just finished reading Brotherhood of Kings by Amanda Podany and apparently Amenhotep II stated outright that "from time immemorial no daughter of the King of Egypt is given to anyone."
Do we know why this attitude, so different from that of the nearby Mesopotamian kings, evolved in Egyptian society? What did the princesses do if they weren't allowed to engage in diplomatic marriages? I know that some married their brothers in order to preserve the royal line and some ruled outright, but what about the rest of them? Did they serve the Temples? Doing what? Or did they exclusively focus on childrearing?
NOTE: I posted this last night and got 10 upvotes but I had accidentally typo'd it as "Priestesses" thanks to autocorrect :( Sorry for anyone who was curious about Egyptian Priestesses (who are super cool, but pretty well-covered by Ancient History Encyclopedia).
Not to preclude further answers, but u/ianwill93 wrote about what little we know about the lives of Egyptian princesses in I'm a Royal Princess in Ancient Egypt. What Do I Do with My Time?
To add a bit to his post, princesses of the Old and Middle Kingdom were often married off to high officials, which cemented ties between the royal family and key officials, particularly those outside the capital. For example, Penn's excavations at Abydos turned up seal impressions of the king's daughter Reniseneb, who seems to have been a 13th Dynasty princess married to the mayor of the region.
By the New Kingdom, the Egyptians frowned on princesses marrying anyone other than their (half-)siblings, and most remained unmarried. This prohibition did not apply to women attached to the royal family through the queen. For example, it is often assumed – though not proven conclusively – that Mutnedjmet, the wife of King Horemheb of the 18th Dynasty, is the same individual as Mutbenret/Mutnedjmet, the sister of Nefertiti, the wife of Akhenaten (one of Horemheb's predecessors).
As you noted in your post, the Amarna letter EA 4, in which the king of Babylonia (Kadašman-Enlil I?) writes to the king of Egypt, has often been quoted in this context. It is always hazardous to take such claims at face value, however, and I think we should be very cautious about assuming that no Egyptian princess ever married abroad. Some have argued, for instance, that the young woman in Egyptian garb depicted on a vase of a king of Ugarit (a major city in what is now Syria) is a depiction of an Egyptian princess sent to the city as part of a diplomatic marriage. For more on this, see Marian Feldman's "Ambiguous Identities: The 'Marriage' Vase of Niqmaddu II and the Elusive Egyptian Princess." It is also worth noting that the Hebrew Bible/Old Testament references Solomon marrying a daughter of the king of Egypt (1 Kings 3:1), which would have been about 400 years after the time of Amenhotep III. It is not at all certain that this marriage took place or even that Solomon existed, but it at least suggests that such marriage alliances were not unheard of.
Some great answers about Egypt from u/Bentresh, so this archaeology trained high school teacher will speak to what I'm more familiar with - Ancient Mesopotamia and the rest of the near east!
It's actually serendipitous that your autocorrect changed it to 'priestess', as in an interestingly similar vein to Carolingian princesses as abbesses millennia later, it seems that some princesses, especially in the Early Bronze Age, served roles as priestesses or at the very least patrons of important cults in important centres like Uruk. In fact, the woman who is sometimes considered the first attested and named 'author' - En-hedu-ana, served as the priestess of Inanna (who became connected to Akkadian Ishtar) in one of the oldest temple centres at Ur. She was the daughter of Sargon the Great of Akkad, and through her hymns composed for various temple centres across the Akkadian Empire we learn she held an important position in uniting the various cult centres, and was even expelled from Babylonian Ur to her home in Akkad before being reinstated by her father. What is most important to this context with Enheduanna is the first part of her name - EN - is an important title often given to high priests and forming the beginning of the gods Enki and Enlil - the latter of whom is the god of the Sumerian Flood Myth!
From the Early Bronze Age Enheduanna (ca 2300 BCE) that title of EN is seen in the names of some princesses of Babylon, the Neo-Babylonians, and the Neo-Assyrians, but sadly we have very little surviving record of them beyond their names. However, nearly 1600 years later in the 7th century CE, the sister of Assyrian King Ashurbanipal, Serua-eterat demonstrates more agency. Some of her writing survives, and the most complete of her writings is a letter to her sister-in-law, Liballi-sharrat, Ashurbanipal's queen, admonishing her for not practicing her writing:
*"*Why do you not write on your clay tablet? (Why) do you not rehearse your exercise tablet? Otherwise they will say : "Is this the sister of Šeru’a-eṭirat, the eldest daughter of the Succession Palace of Aššur-etel-ilani-mukinni, the great king, mighty king, king of the world, king of Assyria?" And you are the daughter, the daughter-in-law - the lady of the House of Assurbanipal, the great crown prince of the Succession Palace of Esarhaddon, king of Assyria" [Serua-eterat's father]
Beyond the incredible ancient listing of titles, Serua-eterat's letter has some interesting implications: first, it implies that literacy among Assyrian princesses and noblewomen was not only common, but an expectation; secondly, almost paralleling Enheduanna, she reiterates her position as eldest daughter of Esarhaddon, and moreover the older sister of Liballi-sharrat's own husband. In other noblewomen's writing from some of the libraries and archives in Mesopotamia we even find them admonishing men for writing poorly. Therefore, there's the tantalizing possibility of princesses and other high status women having privileged access to the court and other officials, and perhaps even serving in those roles themselves.
We know that a few centuries later via both Herodotus and other textual evidence that there are women with not only a right to own property, but conducting their own business and perhaps even holding court appointments - such as the businesswoman Irdabama in the reign of Darius I of Persia. Digressing a little bit, this ability to own property comes back to the roles of priestesses like Enheduanna, as seemingly as far back as Akkad and at least Hammurabi in Babylon we know of the role of naditu from his eponymous code. These naditu were women, sometimes slaves, who were entitled to own property themselves while living in a temple complex - while they may have lived communally like European convents much later, archaeological evidence indicates that at least some of them lived independently within the complex. Returning to Enheduanna, from some of her letters it shows her supporting slave women, temple prostitutes, and priestesses living within the temples throughout Mesopotamia - demonstrating at least some independence and status for the daughters of Mesopotamian kings.
On the periphery of the Near East among the Hittites and Hurrians are some less known but interesting cases of 'princesses' holding power beyond priesthood or court roles, as a great part of the Hurro-Hittite corpuses we have today are invocations for pregnant women brought by female members of the royal families almost as often as their own families. These invocations of female underworld deities also extend to requests for protection and expulsion of demons and illness, often on behalf of a high status woman's name alone.
It is in the Kingdom of Mitanni where it comes full circle back to Amenhotep III and the Amarna letter you mention. The Hurrians of Mitanni in what is now Northern Syria participated in the Late Bronze Age networks of royal gift giving and exchange that are exemplified in the Tell-el-Amarna letters and the Hittite-Egyptian Peace Treaty. The contemporary king of Mitanni, Tashrutta, gave his siter Gilukhipa, and his daughter, Tadukhipa, as wives to Amenhotep III - along with 317 concubines! As u/ianwill93 mentions in his response about Egyptian princesses that u/Bentresh introduced me to here - these princesses were but more along with Babylonian and other near eastern princesses already - there are even some scholars who believe that Tadukhipa, who married Akenaten after Amenhotep's death, is either Nefertiti or even Kiye, Tutankhamun's likely mother. However, though he claims no Egyptian daughter has ever been married to a foreign ruler, two centuries later, a widowed queen of the Hittites petitioned the Pharoah for a husband for fear of being killed and was seemingly sent an Egyptian-Hittite nobleman, almost ironically doing the inverse of Amenhotep's letter.
As you note there does seem to be a distinction between the role of princesses and daughters of the Pharoanic family and those of Babylonian and other near eastern rulers, and this might be tied to a distinction in the role of the ruling dynasties as divinely appointed, as in the case of the Neo-Assyrians, rather than being themselves 'living gods' in the case of Egypt.
All told, it's a fascinatingly complex issue complicated by the lack of epigraphical and archaeological evidence of the roles of royal women, and women more generally. For further reading and some of my sources: