Whether people are referring to רבינו בחיי בן אשר or רבינו בחיי אבן פקודה, you will always find people who pronounce their names as either Bachya or Bechaye (Even Wikipedia uses both!). Is there any source which indicates which version is the correct pronunciation, or perhaps a third possibility? To me it seems as though Bechaye would be the correct pronunciation, based on the spelling of the word (such as ובחיי דכל בית ישראל in the Kaddish), but I have yet to see anyone clearly express how to correctly pronounce it.
In the introduction of the Chovot Halevavot published by Mosad Harav Kook, he brings that it is not known how to pronounce the name. He says that Ashkenazim pronounce it Bechaye, and Sefardim pronounce it Bachye.
See there for some other opinions and rationals as well.
R. Reuven Margalios in his פנינים ומרגליות has a note on this name. His theory is that it is not an actual name, but a nickname which means "may he live long," like the (Yiddish) name "Alter." He proves that it is not a real name because in a manuscript of Kad ha-Kemach, the author is listed as R. Yehuda. According to this, it would presumably be pronounced "Bechaye," since it is a word meaning "with life." The Hebrew Wikipedia entry also quotes a scholar who supports this pronunciation, as it is likely a shortened form of "Bechaye'el."