I just saw a post on Imgur claiming that James I commissioned a new translation of the Bible in order to appease the church, who were aware and critical of his same-sex relationships. While it seems to be commonly accepted that James I did have sexual/romantic relationships with other men, I'm not finding much to suggest that he commissioned a new translation for this reason - his personal religious scholarship and the divisions between Puritans, Protestants, and the Church of England are commonly cited. Are there more in-depth sources that point to church scrutiny driving his commission of the new translation?
When the KJB turned 400 in 2011, there were a plethora of books written about its translation. Gordon Campbell's Bible is probably one of the best.
The Authorised Version came about after the Hampton Court Conference of 1603/4, which convened to address concerns of the Millenary Petition. By most accounts, John Reynolds (Rainolds) was one of the two puritans in attendance and made an off-hand comment about needing a new translation of the Bible. King James ran with it and formed what might be the most bureaucratic and academically intensive translation to date (in terms of contemporary knowledge of language with available translations; newer translations are certainly more academic and accurate to the originals) with 3 committees (each with two subcommittees) of about 60 translators. The translators even came up with a list of rules on how to approach the new translation -- several of which are fairly conservative in terms of preserving older translations and religious vocabulary (so the "third translation from the Hebrew" is a little misleading; the Hebrew was consulted, but they still favored previous English translations). It's been a few years since I've read a lot of the literature on the translations, but I don't recall reading anything about James's personal life playing into the commission for a new translation -- though James certainly thought himself learned in theology as his publishing record attests.
Particularly poignant is that the commission occurred in 1604, the Bible was published in 1611, and James met Villiers in 1614. I'm not saying that James didn't have other male lovers, but I think there's a bit of playing with chronology to revise the history of the KJB.