If the original screw was 12mm, the carbide drill remnant can't be very long or deep in the hole. I would try a hardened pick to break the drill. I had this happen when installing a cover for a DRO on the back of my lathe. I was able to break enough of the drill to use a shortened screw.
There are diamond bits for a Dremel that will cut the carbide although you may go through a few in the process. There were a number of threads on this site regarding home built sinker EDM machines.
The approach depends a lot on whether the item is small enough to work on a mill or drill press, if it is cosmetic, how soon does the item need to be back in service, etc. Personally, I would try the pick approach first. An old chainsaw file can be ground into a tiny punch and used to break the carbide. Try to hit the drill flute rather than dead on. If you resort to the diamond burrs, go slowly to allow the diamonds to do their work. It is easy to strip them off the burr, rendering it useless. As a last resort, I would consider the home built EDM.