The shelf mat'l is sinew?? First time I've heard of using sinew for that.[:-] Somebody must've had more sinew than he needed to use it like that. But that's beside the point.

If it's sinew, I'd leave it be and wrap over the top of it. Most string making mat'ls are coated with wax. If you intend to use it, then get as much of the wax off as you can before using it. Upholstery thread - any fabric store will have it - or braided dacron fishing line will do both a good job and you won't have to worry about wax.

Just be aware that wrapping over the sinew and then applying another shelf covering will change your degree of centershot. In that pic there, it looks like that sight window is cut practically to center anyway. I don't like to critique other's bows, but I think that's cut way too deep. Probably what weakened it in the first place.