Might I suggest you consider one of the easiest backings out there....rawhide.
If that were my osage board, I would use that line you drew as my center line, then define the sides of the bow accordingly. I would add a very slight crown, no more than about a 1/16" because I like the aesthetics. Before I even started going for floor tiller, I would radius the corners on the back and lay down my rawhide using Titebond (matters not which one, but III has some benefit of being a moisture barrier). For giggles and camo, I might even use a combination of aniline dies (black, green, yellow, red, blue) to paint a random pattern before laying on the rawhide....thin rawhide from a deer or pronghorn antelope will gently mute the brightness of the aniline dye and leave a nice camouflage effect.
Don't go the route of the rawhide dog chew toys for backing, one never knows how thick it is or how it was treated before it got to making a chew. For that matter, I wouldn't feed it to a dog for the same danged reason! If you need help sourcing rawhide strips for the backing, drop me a note. I have some ready to go.