We spent a week on Ambergris Caye last year late Aug/early Sept. It only rained during daylight hours for about an hour the entire week! Contrast that to the two weeks we spent there in May where it rained 11" in one day...not sure about this rainy/dry season thing.
It rains much less in the north of Belize than in the south, and less on the Cayes than on the mainland. What we experienced was that most of the rain came through at night. I wouldn't let rainy season stop you from visiting Belize, at least the northern part.