So, I agree that you should just try and chase a ring in the sapwood. It definitely looks violated, but the bow would look great if you could keep some sapwood on it.
Now I have a question for those with more experience. A couple of you have said it should be relatively easy to chase a ring in Elm. I've only ever chased a ring on Osage, and that was easy to do with my drawknife. For those with more experience, how would you chase a ring on Elm? I'd imagine it would be difficult with a drawknife. Would you rasp it? Or maybe use a scraper?