Yeah, that stinks, and they should be a little more flexible if it was indeed their mistake, but you also need to take this life lesson and learn from it. I have auto-pay set up on many accounts as well, but I check them and confirm the payment each month anyway. To me, auto pay isn't so much a convenience feature that I would rely on, it's a safety measure in case I get busy or absent minded one month and forget to check an account. The likelihood of both me and the payee "screwing up" in the same month is pretty small.
I understand that time is critical here, but if you want it off your report(s) even after the mortgage (assuming you can't get it off now), you may have success down the road with a goodwill letter.
Another option is doing some research into dealing with corporate call centers and try again. A few quick tips:
1. Keep asking to escalate the call if you don't like the answer (ask for their supervisor/manager)
2. Try back another time - a different person on the other end of the phone may be able/willing to help you even if the first person wasn't. Heck, even the same person on a different day may give you different options. Persistence pays off, as debt collectors well know.
3. Be civil with the person on the other end of the phone. It's not their fault that their company screwed up or what the policies are. They're also more likely to want to help you if you treat them like a friend you're confiding a problem in rather than the enemy that caused the problem.