Is this a motor mount, or a simple stud set into a hard point?
Usually for a broken stud, you would either arc weld a bar to it, and then spin it out, or drill down into it (with a hardened metal bit) and then insert an "Easy out", which screws in and then screws the stud out. (They're reverse threaded.)
Could you get in there with a right-angle drill? A right-angle adapter? A Dremel with the flex adapter on it?
You could probably
rig a saddle over the mounting foot to hold it down, as long as there's enough stud to keep it centered properly...a kludge job but one that could be perfectly good if inelegant.