Kalaehina:
I love that idea! My breakoff screw was really tough also. I tried the torques bit route, and had no luck. What i ended up doing was cutting a hacksaw blade off and useing it to cut a slot in the top of the screw for a screwdriver blade, then I tapped on the screwdriver while turning and it came right out. I think the paint had it more than anything. I also believe you could cut the head completely off then take the screw out after the cover was off, just don't want to drop it down the hole!
Stick