Since you have to wet the rawhide for applying it's not a good idea to paint on it's backside before the gluing. The fleshside is not as good a paining ground as the hair side, also.
I've done several bows with paintings on a rawhide backing with both acrylic and India ink. see below.
Your plan could work as that: stain the bowback dark first, put on your rawhide, thin your India ink with water and it'll pretty much have the effect you seem to want!
Or:
Paint acryilic on the back of the bow, steelwool it and then put on the rawhide; but rawhide should be sanded very thin, like pergament (I used goat pergament on several bows; I believe it's the best choise since it is very fine, VERY strong and very even in thickness AND it's great painting/drawing/writing ground)
Hope this helps; I love painted bows!
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