Okay, so Amex refused the claim.

Reason:

"we are unable to approve it because this property is an all-inclusive in which is not eligible for the program."

Which is really odd because:

1. The hotel is exclusively all-inclusive. Regardless of where it's booked.
2. It does have the 'lowest rate guarantee' badge.

Naturally, I refunded the reservation, and then saw this:

"Please note your refund may take up to 45 days to be fully processed."

45 days? Yikes!