Some things that come to mind... how were the gluing surfaces of the wood and sturgeon skin prepped? Sturgeon skin is oily and should be degreased. Did you do that? How? After dying the back of the bow, did you allow it plenty of time to dry? Did you check for and remove excess dye residue? Was the skin dry, wet, or damp when you glued it on? Did you size both surfaces with glue and allow it to soak in and dry for a while before applying a fresh coat of glue to both pieces again just prior to laying the skin on? Was there excess glue squeezing out the sides as you worked the skin down against the wood? Did you wrap it? If you didn't wrap it, how much extra skin was sticking out past the sides? How long did you let it dry? How and when did it come loose? As it dried? Afterwards, when the bow was drawn? Keep asking yourself those kinds of questions and the answer might come to you.
The glue may not have been the problem in your case, but I prefer hide glue for sturgeon skin.