Another option, if it isn't countersunk, would be to file two opposite sides of the head flat and take it out with an open end wrench or vice-grips.
Re: Welding: A good way to weld to a cap screw is to sacrafice an allen wrench. Put the wrench in the stripped hole and weld it there. This way you don't have to get "behind" or underneath a nut. It takes less welding, too (brazing works too if the area can take the heat, or if the cap screw doesn't go to ground)
Re: Drilling: Forget it. Most cap screws are hardened and drilling them is a nightmare, unless you can heat them up first.
1982 Mercedes-Benz 300CD
1982 Mercedes-Benz 240D - stick