I do like Joe does. If you add a thin layer of TBIII to the bow first and let it dry completely then add the rawhide, with glue on the bow and the rawhide, the first layer of glue will act like a vapor barrier to prevent the moisture from the glue and rawhide from entering the wood.
I had an osage bow loose 10# after adding a rawhide backing. I had to wait 2 weeks before the draw weight got back up to where it was before.