It doesn't take much wood to make a bow. I would definitely get the sapwood off those staves and seal the backs really well with shellac. I band saw most of the sapwood off and clean up the backs with a draw knife.
You could lay out your bows, cut them a little wide and heat correct any deflex, twist and dogleg.
If it was me, after I got my mulberry bow blanks made I would put them aside for more curing and get that huge osage taken care of before the bugs carry it away. If it has powder post beetles in it you don't have long to take care of it before they ruin all the wood.
I went to a friend's in-laws farm once to look at all the osage they had pushed over, at least 100 trees, I thought it was going to be my lucky day. The trees had been on the ground for a year or so and every one had powder post beetle holes going in the top of the trunks and out the bottom, all was lost.
Getting the tree cut up, split, bark and sapwood off and backs and ends sealed will prevent the above tragedy. There are a lot of staves in your tree but you will be surprised at how much waste you will have as well. Every piece of osage has a surprise under the bark, some good but most often bad.