I wouldn't worry about the mosquitos on Ambergris Caye. It is rare that anyone is much bothered by them in town or on/near the water anytime of year. Only on those times, usually later in summer, when the prevailing breezes die down are they a concern on the beach side. Yes, on the back side of the island away from the breeze they can be vicious, especially in the far south and north, but again you likely won't be spending much if any time where the mozzies are.
As to the rain, unless you are very unlucky you'll have plenty of sunshine and good weather most of the time. There is no "monsoon season" in Belize -- just seasonal rains, but on the cayes unless there's a stalled tropical front or storm the rains come and go pretty quickly.
On hotels in the US$150-$200 range, that's really plenty off-season. Most hotels and condotels are less than that, especially if you book direct on the Internet.
Among the places I like (most less than your budget) are:
**** Banana Beach
**** Caribbean Villas
**** Banyan Bay
**** Victoria House
*** Mayan Princess
*** Paradise Villas
**** The Palms
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