From a machinist's point of view, here's what I'd do. Get a center punch (they're only a couple of bucks) and carefully center-punch the screw. This will produce a small divot that will keep the twist-drill ("bits" are for wood working) from "walking". Next buy the proper-sized twist-drill and ez out--again, only a few bucks. Then drill and ez out the screw. Patience and penetrating oil are your friends. Then buy the proper tap-again only a few bucks- and re-tap the hole. You should be able to fix it for less than $15. You'll have a few new tools for your tool box and the satisfaction of having done it yourself.
Please keep us posted on the out come.
Jon