I will assume that stripped means the bolt (not a screw) is frozen/rusted in place and you have rounded the head. If that is what you are talking about, getting it out on the bench to be removed/drilled out is the way to go. I do not know how hard you will have to pry/hit in order to snap it. I think you may break something, including the glass. But a steady force may just break it off. You can gauge if something is getting too stressed inside there. But again, I don't know what is "stripped"? If the head is rounded, there is no bite for a socket.
Cutting a hole "seems" like a big deal but it is not.. When you cut the access hole, Just choose good cut lines on flat surfaces even if you have to cut larger than necessary. Weld/grind/smoothing will be easier on flat than on a dimple, crease, ridge, transition step.....
Cut a slot in head, or any method you can thing of, after you cut the access hole. Try real hard to get the whole bolt out before cutting off the head. There is a method of welding a nut to the head for a new grip. This is after soaking with penetrant.
You may want to take the door off to do the cut and weld. Laying flat on a table is easier and will give a better work position. Laying flat allows gravity to evenly soak threads. This is in keeping with Brad Allen's idea of taking the problem to the bench. I hope we are talking cutting a hole on the inside, not the outside of door.
After you are done, no one will ever see the cutout.
