Part of your problem is the rawhide you're using all wrinkled and warped like that from being all bent and twisted in the dog chew. You are fighting the warp and ripples the whole time. What I have found to prevent that is to soak the rawhide till it's good and flexible then stretch it and staple it to a flat piece of plywood. Get it as flat and tight as possible, pulling out all the wrinkles. Just like if you were preparing the hide fresh off the animal. Let it dry till hard. If there are still small ripples you can repeat the process (much easier the second time). Once it is dry and flat, you won't be wrestling the ripples as you are trying to put it on the bow. And like Blackhawk said, trace and trim it close to the final size and you won't have so much to deal with
And to your comment about the glue not sticking to the hide, Loneviking, if you are still using tightbond, it's not going to have the same affinity for the hide that hide glue will. Hide glue and the hide are basically the same material so they will stick much better to each other whereas trying to coat wet/damp hide with tightbond, you are thinning the tightbond with the moisture in the hide. Maybe the guys who use it can give us an idea how to deal with that.
Lots of great tips on here I'll be using in the future. Just shows ya there's many ways to skin a cat - or a bow in this case - sorry Del ;-)