Unless you can point to a contract provision which guarantees only one fitting in a time-frame, you are going to have a very tough time on this one. It's hardly usual for there to be only a single fitting, so it's not as though the vendor is doing anything out of the ordinary.
It won't hurt you to make the case to Amex, but you can expect the merchant vendor to fight this one tooth and nail.
Isn't the better approach to try to work this out with the vendor and find times which might work? Maybe speak with someone else.