If you're desperate to fix it and can't pursue the other options, here's what I would try... but test it on some scrap first. Drill a hole about the diameter (or slightly larger) of the broken screw hard up against it on the inboard side. Then use an awl or screwdriver etc to pry/lever the broken stub sideways to the new hole to release it and get it out. Mix some sawdust with your blue stain and when dry mix in a touch of Titebond and work it into a firm putty, hopefully matching the body colour, and plug the hole (should have about 5mins working time with the glue). If the mix is firm enough you should be able push it in flush with the surface and not need to sand. Give it 24hrs, drill your new hole, and you should be good to go. I haven't tried this, so I'd test it on some scrap wood first (screw in a small screw, cut the head off and give it a try). Caveat emptor.