In general, the food is safe, and the tap water in most cities and towns is potable (I live here). Outside cities & towns, you might want to check in case the water is from a well or rain-water tank.
However, I wouldn't rate the hygiene standards very high, especially if you're eating "low budget". Standards are probably very high in the bigger hotels and higher priced restaurants, but for the rest, standards are probably worse than your own kitchen.
HACCP has been introduced in Belize (within the past few of years, and I know a kitchen worker who has attended a course) but not all establishments follow it.
In speaking with a local doctor last year, he said that intestinal bugs & parasites are common in Belize (from untreated water, raw/unwashed fruits & vegetables, poor food handling techniques especially with chicken & pork). Some of the parasites do not present symptoms and the only thing to do, to be sure, is have a stool sample analyzed.
The health department often inspects kitchens. They did one last week after a customer found rat droppings in a loaf of bread.
I often eat "low budget" and have only had "food poisoning" in Australia! Even survived the old lady in Mexico making my garnaches with her bare hands.
Sounds like you know some common sense, so you should be fine.
Edited by: misterbee