I would guess that your easiest solution will be to get a court order awarding you the house in the divorce - which should take the place of the quit claim deed.
He could easily argue that he's entitled to 1/2 of the marital equity, but since there would have been almost no increase in equity during the marriage, you'd be fighting over an insignificant figure. I'd personally come up with some way to estimate the increase in equity over the marriage (probably by assuming that the home value didn't go up, and then using the decrease in mortgage balance as the equity gain), and offer 1/2 in the settlement proposed to the court. You might be able to argue that he's not entitled to any of it, but that's probably a losing battle - and would cost more than you'd save.
Unfortunately, if he won't sign a quit claim deed, there's nothing that you can possibly do to resolve the issue in 2 days. ANYTHING that you might do is going to take much longer than that.