I've been going to Belize for years, almost always in June. It's the low season so there are fewer crowds, which is a big plus for me.
The rain in Belize is really no big deal. In the summer, it's almost always at night and sometimes with some really terrific thunder and lightning. The rain tends to taper off and stop in the morning. By mid-morning you can't even tell it rained the night before.
I've had very few dives called off because of rain in the summer. Wind is a much bigger factor since it effects the waves.
Go in June. You'll love Hamanasi. And while they have a very good restaurant, be sure to also check out Chef Rob's Cafe a little further down the road as well as the sandwiches and smoothies across the road at the Oasis Gym & Cafe.
Happy Anniversary!