It definitely sounds like he's a very stressed out little guy
if a tank style cage has decent ventilation and it's kept in a cool, dim corner away from direct sunlight then it won't necessarily get hot. Most tanks can be either left open if they're tall enough or you can make a mesh lid that prevents escapes but allows for maximum air flow. Bin cages can have all 4 sides and the lid meshed if you have the patience so they have as much air flow as a barred cage and they are always the cheapest option. If you're worried about chewing mesh (my cheeky girl still finds a way to make noise with mesh and get my attention!) then just mesh the lid!
I hope you don't decide to give him away, you haven't reached the end of the reasonable solutions yet!
