Is there a special word for determining who wrote what part of a book, e.g., the Bible?
The field of study you are referring to is called "authorship attribution."
So you can work backward from that to what you need, e.g., authorial attributor," etc.
The program, part of a sub-field of artificial intelligence studies known as authorship attribution, has a range of potential applications — from helping law enforcement to developing computer programs for writers.
The general activity is usually called source criticism, though Wikipedia lists related terms
- cognitive authority; authority (textual criticism)
- credibility (e.g. media credibility)
- critical literacy /critical reading /critical thinking /Information literacy
- information criticism /information quality /information evaluating
- quality of evidence / quality norms in science and scholarship
- source evaluation / source reliability
- trust (social sciences); trustworthiness
Although I believe it to be a singularly awesome book, I think the Bible is a bad example for this question because those who take part in studying it cannot easily be labeled with single term such as you are looking for. Its history is much longer and more complex than any other work and unraveling the chain of revisions often falls to theologians as much as to linguists or historians. No other book has been subjected to so much translation, replication or preservation. And most importantly, it's claim of divine inspiration completely changes the rules for whatever "field" attempts to study it.
In addition to The Raven's answer about authorship attribution to cover the field in general, it might be worth noting some of the sub-fields that go into this. One of those often used in identifying ancient texts is Stylometry. Of course that's only one aspect, and the various people that contribute ideas and research when attributing ancient texts are usually referred to by their more specific realms of study rather than by an umbrella term like you are looking for. Historial linguists, historians, archaeologists, various time period or cultural expercts, and in your example theologians all play roles.
Don Foster is an English professor who has uncovered unknown authors through textual analysis, and has been called a "forensic linguist" who practices "literary forensics".