He could be urinating all over himself if he’s drunk more/licked ice so his bladder gets full.
Wet their ears with cool water. It helps them cool down. Can you get a fan of the type that you can hang a wet towel over? That is a very effective way of cooling a bunny down, if you place the fan rather near the rabbit (beware of cable chewing).
Cool water bottles are good as well as bricks/tiles/ stones that you’ve kept in the fridge or freezer for a while -exchange often. Buns often like to lie close to or on the cool objects.
As for fly strike, clean Gary’s skin free of urine and put a barrier crime on his downstairs area (vaseline) to keep him from getting urine scald/compromised skin -that makes him less attractive to flies. Check butts twice a day for signs of flystrike if you see flies near them.
For humans it’s good to wrap them in wet bed sheets and put a fan on them, but I don’t know how well that’d work on a bunny.