Hilton has the ability to bill the hotel through Guest Assistance for compensation offered to a guest when the hotel refuses to play ball. If the hotel insists on charging the higher rate, it's entirely possible for Hilton to refund the guest directly and then bill the hotel for it.

I would not be surprised if they end up doing this if the hotel doesn't blink. Assuming everything the OP said is accurate (and I have no reason to believe it isn't), this is exactly the kind of case Hilton would do it for.