I would think drilling it out would be your best bet. StewMac sell a screw removal kit for this type of job. It’s expensive though. It’s easy enough to get a dremel, drill small holes in a circle as close to the broken screw as you can and then slowly go round the circle of holes with the dremel until you can remove the broken screw. Plug with dowel and a dab of wood glue , colour match and replace bridge hiding fix.
Stainless steel replacement screws are a good shout. I had to do the above fix on a sheared machine head screw. I was missing one machine head screw after replacing the machine heads on a bass. All the’70s screws went in fine. The modern replacement screw sheared off. As someone said made of butter.