I find it much more likely that the rate OP was charged was the rate on the reservation/confirmation and the hotel just never got around to responding to the inquiry from AmEx or, more likely, decided it wasn't worth the customer service impact for a number that's very small to them. Quite often, if you whine loudly enough, a company won't bother, even when they're right. I suspect OP just got away with $182.

Even with Marriott's integration IT issues, this is core functionality that has existed for a long time and operates thousands of times a day. The odds of spontaneously generating a new daily rate on a reservation is quite high, IMHO.