Do you know for sure how the stave was treated after it was cut? If so you can leave the sapwood on. If you are unsure, DON'T!
One option for a stave that is thin in the center and thicker on the outside ends is to cut it in half and splice it back with the limb tips being the thin ends. You could also make a bend through the handle bow with it or just add a handle riser and go from there.