Thsese are a genuine questions, because I have not the foggiest notion of the answer. If you were conducting experiements at odds with Mormon doctrine, would the be alloowed? If the results of your experiments offered strong evidence disproving some aspect of Mormon teachings that made empirical claims, would those results be acknowledged and would your position be safe?
State universities have a long tradition of academic freedom, the tradition among religious universities is mixed. I do not claim to know where BYU falls on that scale.
I worked for 4 years for a state university in Wisconsin, and I've worked for 4 years here at BYU now. From my perspective, the academic freedom is identical. Of course, I'm doing physics research and not sociological, so perhaps I just haven't stumbled into any problematic areas. But based on conversations I've had with colleagues who are doing things that are potentially more closely related to religion, I would say even in those areas the academic freedom here is as large as it was in Wisconsin. Different, in a way that I'll explain in a second, but just as large.
One difference is that one actually HAS the freedom to pursue religiously-related research here at BYU. In Wisconsin, as near as I could tell, religion was taboo. There's no way that anyone would be involved with anything like this religious documents project there, for example:
https://cpart.byu.edu/. So for that type of thing there is actually MORE academic freedom here than there. However, as you have correctly surmised, there may be LESS academic freedom at BYU in some other respects. If you start teaching as fact things that are contrary to official LDS doctrine, then I'm guessing the university would have a problem; that's apparently what happened back in Sept of 1993. (See
https://en.wikipedia.org/wiki/September_Six.) So those two things average out, to my way of thinking.
The other thing to note is that the LDS church has consistently taught that all truth is part of the gospel. See
Teachings of Brigham Young, for example, in the section entitled, "The gospel of Jesus Christ embraces all truth.".
Quotes from Brigham Young said:
All truth is for the salvation of the children of men—for the benefit and learning—for their furtherance in the principles of divine knowledge; and divine knowledge is any matter of fact—truth; and all truth pertains to divinity (DBY, 11).
“Mormonism,” so-called, embraces every principle pertaining to life and salvation, for time and eternity. No matter who has it. If the infidel has got truth it belongs to “Mormonism.” The truth and sound doctrine possessed by the sectarian world, and they have a great deal, all belong to this Church. As for their morality, many of them are, morally, just as good as we are. (DBY, 3).
Such a plan incorporates every system of true doctrine on the earth, whether it be ecclesiastical, moral, philosophical, or civil; it incorporates all good laws that have been made from the days of Adam until now; it swallows up the laws of nations, for it exceeds them all in knowledge and purity, it circumscribes the doctrines of the day, and takes from the right and the left, and brings all truth together in one system, and leaves the chaff to be scattered hither and thither (DBY, 3–4).
All knowledge and wisdom and every good that the heart of man can desire is within the circuit and circle of the faith we have embraced (DBY, 446).
It embraces every fact there is in the heavens and in the heaven of heavens—every fact there is upon the surface of the earth, in the bowels of the earth, and in the starry heavens (DBY, 448).
Our religion is simply the truth. It is all said in this one expression—it embraces all truth, wherever found, in all the works of God and man that are visible or invisible to mortal eye (DBY, 2).
etc.
Therefore the church teaches that if there is an apparent conflict between science and religion, it's either because we don't understand the science well enough, or we don't understand the religion well enough. That's why there are people here at BYU who study evolution, the big bang, and so forth, even though other religions may feel that there's a conflict between the Bible and those scientific theories. And that's why people at BYU were free to study ancient Mesoamerica and conclude that it's likely the Nephites/Lamanites were NOT the sole inhabitants of the Americas when they settled. And people at BYU are free to study DNA evidence which may lead to the same conclusion. As discussed in another recent LDS-related thread here, what "gave" in that apparent conflict between science and religion, was the notion that the Nephites/Lamanites were the "principal ancestors of the American Indians" (as the BoM introduction used to read). In that discussion I pointed out that the Introduction to the Book of Mormon now reads just that they were "among the ancestors of the American Indians". So faculty at BYU are absolutely free to apply scientific rigor to any of those types of things. Now, where the limitation on academic freedom would come into play, would be if someone researching Mesoamerica had said "There is evidence of multiple cultures present in the Americas during and before the time of the Nephites. That proves the Book of Mormon is false." instead of saying, "There is evidence of multiple cultures present in the Americas during and before the time of the Nephites. That proves we should reexamine our assumptions about what the Book of Mormon teaches," as they in fact did. (I think that was John Sorenson from BYU's anthropology department who said that type of thing 20+ years ago, but I could be misremembering.)
Probably a longer answer than you wanted, but that's my 2 cents about academic freedom.
