If I had adequate thickness remaining yet, I would first try to chase a single ring end to end. What do you have to lose? So you may have to back it? Big deal, woulda had to do it anyway. Put the drawknife away though
If chasing the ring didn't work out, which material I used to back the bow with would at least partially be determined by other design features of the bow I had in mind. If it was shorter, narrow, reflexed, recurved, had any descrepencies like knots in the back, or runnout, or otherwise pushing the woods limits, I'd use sinew. If it was long, wide, straight, and cleaner, etc. I'd use rawhide.