I am from Texas and like most am addicted to A/C when it's hot. My first trip to Caye Caulker was in August. It did take some adjusting. We had a three bedroom house with window units in two of the bedrooms. During the day try and leave the units on low to cut down the humidity. You will be out and about, don't overdress. Take shorts, tshits, and flip flops, won't need much else.
The restaurants have no air con, but most are near the beach and breeze. You will adjust within a couple days hopefully. The other trips I have taken were earlier in the year and never had a problem with sleeping. Make sure you have at last one A/C unit though. There is plenty of shade on the beach with palm trees if you need it. You will have fun, just keep your expectations simple. It's a very unique place and is hard to beat!
Hope to be going slow soon!