I use the dog chewy rawhide often, as I am too cheap to purchase the "good stuff", although, IMHO, the good stuff is the same as dog chewy stuff, but the dog chewy must first be worked and thinned down to an acceptable thickness. I usually soak it, open the knots, and then tack it down to a piece of ply using finish nails, let dry, scrap and sand, wet again, tack down again, scrap and sand again, and usually by this time, it is good for use on one of my bows. When thinned like I describe, I think you will not have such trouble with it laying flat.
I pretty much do the backing job similar to bluegill, but, once dry, I will actually scrape the hide thinner being careful not to scrap thru (which happens sometimes, but, shhh, don't tell, the stain hides all flaws).
Russ