We built the outdoor rabbitry under oaks and in a shaded area, also use the frozen water bottles.
We have fans to set up to push air through (as you mentioned the humidity!) So at least we can keep the air moving! It definitely gets brutal here June/july & august !

We have fans to set up to push air through (as you mentioned the humidity!) So at least we can keep the air moving! It definitely gets brutal here June/july & august !
I use a fan in the hotter months. I also use frozen bottles and the fan blows on the bottles. The bunnies lie behind the bottles so it is sort of like outside air conditioning. It gets very hot and humid here for months and they have done well with that. Their hutch is also under a big oak tree.