In "Grammar in Early Modern English" by Edmund Weiner, deputy chief editor of the Oxford English Dictionary, I read that the word its, the ubiquitous English third-person possessive of the pronoun it, is a relatively recent innovation that first appeared in print in 1590 and then rapidly took off. Before that precious invention came along, people would use other terms like thereof or his. While Carmack has pointed out that the text is not monolithic and not strictly confined to any one period in time, still, if the Book of Mormon text is primarily influenced by EModE from before the KJV era, then we might expect its to be less common in the BOM than in the KJV. However, the opposite is found. The word its occurs over 40 times in the Earliest Text, but only once in the KJV (Lev. 25:5).
The problem was ameliorated by looking in the OED itself (see OED Online entry on Its) and seeing that its is actually attested even earlier than 1590: 1577 R. Robinson Certain Select Hist. Christian Recreations sig. B.vii, There stands a bedde, its death to tell.
Finding its in print in 1577 gives us more leeway for its use as a legitimate pre-KJV EModE word. Note also that the Book of Mormon still uses thereof much more than its, about four times as often. I suspect that this ratio may be uncharacteristically low for EModE and may represent the intrusion of more modern English into the translation process. Carmack does not argue that the BOM is pure EModE, but argues that it is a complex, non-monolithic mix with a little modern English and a good deal of Early Modern English (see Stanford Carmack, "Why the Oxford English Dictionary (and not Webster’s 1828)," Interpreter: A Journal of Mormon Scripture 15 (2015): 65-77).
1598 J. Florio Worlde of Wordes Spontaneamente, willingly,..of himselfe, of his free will, for its owne sake [1611 of free will or of it's owne sake].
1599 Shakespeare Romeo & Juliet ii. v. 12 The sweetest honey Is loathsome in its owne deliciousnesse.
1603 J. Florio in tr. Montaigne Ess. i. Ep. Ded. sig. A3v, My weaknesse you might bidde doe it's best.
The paucity of its in the KJV is surprising to me. Of course, one could ask why, if the Book of Mormon is based on slavishly imitating KJV language, does it use its at all? But of course, careful analysis of the text has already shown it often departs strongly from the KJV language in many ways. While those departures often include authentic Early Modern English, there are elements of modern English and even a touch of Middle English, as Carmack demonstrates. It is not a simple, monolithic text. As for the word its, its presence could theoretically be consistent with the EModE era but could also be an intrusion of modern English in the text.
Personally, I can accept the idea that there was a base translation of some kind that could be accessed by Joseph Smith in preparing the translation. But since he was obviously willing to make corrections to the dictated text, I suppose he could have used some of his own phrasing at times in the dictation as well. To me, it's impossible to remove human influence in this effort. We see the human influence in the original manuscript, the printer's manuscript, the printed text, and the many revisions to the text that we have today. Why assume the human influence wasn't there as the text was being dictated, even though there was a divine engine driving the whole project? We really don't know how the translation process worked, and need to be open to many difference possibilities as we dig more deeply into the language and other evidences that help highlight the complex origins of a fascinating and divine text that, like all scripture, has gone through many human hands (and lips).