Yes, this. I like this way of thinking outside of the box.I have never tried this, but there are tools to remove rounded bolts that grab onto the bolt head more as you turn... if you find a matching size there may be some success with that. It would let you work on the outside of the fastener if the inside has become unusable....
Ted found the whole kit, too.
I've used these before. Tool stores sell these, most often they are used for removing locking wheel lug bolts, namely the ones that use a key to prevent wheels from being stolen. I've used one of these tools once where I purchased a used car, and the key to the lock lugs was the wrong key pattern.
They are typically made of a very hard tool steel, and thin enough to both grasp the lock lug and fit between the lug and recess it resides in within its wheel:
This could work!