The Bumblebee stud welder is rather nice indeed. We usually don't weld the studs on very tight, and are often able to pull them off or cut them off with sidecutters and reuse them a couple of times. This doesn't put as much heat into the panel, and there's a lot less grinding. It also means that you're not able to pull as hard without the stud coming off, but often that's not a Bad Thing.
Anyway, for the MIG idea, I think you're working too hard. Tack the wire to the panel, shut down the welder, and loosen the wire tension. You shouldn't have any trouble pulling the wire through the gun that way.