This is an old thread but Belize is a great place to go for a quick trip to the Carribean. I went for my honeymoon a few years ago and here is a brief rundown of what I recall:
As far as Belize "foods" go, the unique dishes I recall were: johnny cakes & fry jacks for breakfast. Johnny cakes are sort of a breakfast sandwich made with a sort of biscuit. Fry jacks are fried dough balls, sort of like beignets in New Orleans. Both are awesome.
"Stew chicken" is the uniquitous dish, at least around Belize City, Caye Caulker and Ambergris Caye (the places I went). Typically served with coconut rice & red beans. Coconut tarts are a common dessert. I've been to Tickie's Belizean up near Rogers Park that does a solid rendition of all of these dishes. Make sure you ask for the Marie Sharp's extra hot sauce when you go there.
As far as specific places, there are two places that stick out in my mind in Caye Caulker:
1) Rose's - a small seafood shack just outside of the main strip area. I think it's on the same street that takes you to the water taxi station. I remember asking for the menu and the guy at the grill just pointed to everything on the grill. Can't beat that.
2) Habanero's - This is sort of the upscale place. We had an amazing meal here as well. Worth going to if you want to splurge a bit.
My memory is a little more foggy on Amergris Caye:
1) I think it was Wild Mango's that we went to. I remember having some sort of awesome mexican roast pork dish (cochinita pibil possibly).
2) There is a bakery downtown that had awesome johnny cakes.
3) On the road from downtown back towards Victoria House, there is a small grocery store where I would pick up impossibly thin plantain chips and fry jacks in the morning. I doubt this place even has a name and may not exist any more, but if you bike into town from the resort area, keep your eye out for a little shack on the opposite side of the road from the resorts.