If indeed it is neither on the hotel's property nor is it sponsored by the hotel, and there is no other room, what on earth do you expect the hotel to do.
I would expect the hotel to permit you to cancel the rest of your stay without penalty, but that is the end of it. Simply walking out and leaving it to Amex is taking a significant risk. So long as you know and accept that, go ahead. Me? I'd get this worked out before departure.