Here's a technique I've heard of on TB.
Get some brass tubing with interior diameter just larger than the remaining shaft of the screw. You are going to make a hole saw with it, and remove a plug of wood with the screw shaft in it.
Cut about a 2" long piece of this brass tubing, then file V-shaped teeth into one end. This creates a very small diameter hole saw.
Chuck the tubing up in your drill, set the end of the tubing over the shaft of the screw, and start drilling slowly, my guess is that you'd want to do it in reverse.
You may grab the screw and back it out, but more likely you will drill out a plug of wood with the screw shaft in it.
You then can craft a plug to glue into the hole, then cut the plug off level with the surface of the body. This will give you something to screw into. It may require refinishing if the screw is not covered by some part of the instrument.
I have read multiple reports of success with this approach. It's often used on bridge mounting screws or tuner screws. It does require plugging the hole, but that's MUCH cleaner and will impact less surface area than digging or chiseling out the screw shaft.
Wish I'd though of this! What a great way to solve the problem!
