The water and ice are generally okay on Ambergris. This is not Mexico.
I drink the water straight from the tap most everywhere in Belize (though not in some rural areas ... and in some coastal and caye areas the water may be safe but tastes awful as it's brackish or with a lot of sulfur).
Ditto with ice in drinks.
Having said that, any water that's different from what you are used to may upset your system a little. Also, the combination of heat, intense sun, more alcohol than usual and different foods can also cause problems.
A lot of veteran traveler swear by prophylactic does of Pepto-Bismal (a tablespoon or two a day).