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.
Also want to add that when the system thinks you still have the credit for the year, you'll see popups when trying to book a hotel on Amex Travel which will reference the credit. For this year, the pop up started showing again Jan 1, and after I made a FHR reservation a few weeks ago, the pop up stopped appearing.