I have used Amex at a LOT of hotels in Belize, including some in San Pedro, and at a few restaurants. Also, most car rental companies accept it. Amex is not as widely accepted as Visa or MC in Belize, but much more widely accepted than Discover.
About the Amex gift card, I'm not sure. Probably not widely accepted, and I imagine a lot of places have never even seen one, as they're pretty new.