I don't have personal experience with this so can't guarantee anything, but I would think that unless you cancel ALL of the 2023 FHR reservations, you should get to keep the $200 credit - the terms don't say that you only get $200 for the very first reservation made that year; yes you can only get it once but if you cancel one and you have others, it should apply to the other ones then. If someone has ever had this scenario and the $200 remained on the account then it would be great if they can report it as a useful DP for the rest of us.

If you cancel the reservation to which the $200 credit had been originally applied and it is clawed back by Amex, you do have a compelling argument to get them to reinstate it due to having multiple other reservations. However, it's obviously going to cost you time and energy - I believe you will be successful in the end though but again, no guarantees.