Music wire is usually 1095 carbon steel, which is a water hardening steel in most applications. Really thin sections like knife blades need to be oil quenched to prevent cracking, but that's a special case. The problem with re heat treating the music wire is that you are always going to have a section that got heated hot enough to draw the hardness out but not hard enough to reach critical temperature so that it can reharden. So unless you have a furnace where you can put the whole thing in and start from scratch, you're going to have soft spots.
It shouldn't be necessary to buy a wire bender though. If you have a piece of 1/4 inch or thereabouts rebar or steel rod you can bend the wire around that. The only real worry is putting too tight a radius on the bend which stretches the outside too much.