[KTL]

Just to add to the suggested methods, i've had good luck with welding a "new" bolt to the funked one. I had a countersunk/flat head screw with allen-head wrenching (inhex) as a drain plug on a fancy 3.0L 911 Autospeed oil sump plate that was stripped and overtightened. So I welded the threaded end of a regular hex head screw to the stripped countersunk head. Worked like a charm and no drilling/cutting required.

Point being is that you can get fairly clear access to the shift housing and weld a new bolt on there pretty easy. If you don't have a welder, sorry. Sure you need to protect the interior from weld sparks. But you sorta need to do that if you're drilling.

If the head is not stripped out too bad and has enough meat to support an easy-out, I think the fluted extractor is the way to go. If you can't find one that fits the stripped hole tightly, make it fit. Meaning, you grind the head of the extractor on your bench grinder until it is a tight fit in the stripped head. You should have to tap/press it into the stripped hole a bit. It'll get a good bite and pull that screw out easily.