While there's no tracker like with some of the other credits, the way to tell if the system thinks you have or have not used your FHR/THC credit for the year is to log into Amex Travel and review this info on the landing page:
On the left side, under the image of your card where it says "My Travel Benefits", a reference to the $200 credit will appear each year until you use it, at which point it will no longer show. If you get credit for a booking and later cancel it, if the cancellation is recognized by the system, the reference to the benefit will reappear.
My personal experience (which has been tested many times due to the inability to change dates on existing reservations), is that the system catches cancellations most of the time, but there are the rare times (often during times when people are complaining of other credits not posting reliably) when the system won't catch it.
Is the Amex Travel link you're referring to "https://www.americanexpress.com/en-us/travel/"?
This is what it looks like for me:
https://imgur.com/a/vj9P4Wg