How deep of a hole do you have with the extractor down there?
You may have enough clearance to drill to the next size tap size and cut threads with a bottoming tap. Cut a few good threads and grind the lead threads off the tap to get a few more turns before bottoming onto the broken extractor.
Perhaps a new battery is easier. It comes with less Tylenol.
Crazy me, I would try to get a key locking thread insert in that hole.
Wait....that's a ~$125 battery (Amazon). Buy more Tylenol and go at it!