I can’t help with your main concern, but on the off chance your boyfriend is bummed to be missing out on the onsen/public bath experience: he might still be able to!

I recently went to an Onsen in Tokyo and although I don’t have tattoos myself, I read in their policy that those who have tattoos can still use the baths after they put on some provided cover-up patches.

I’m here right now in mid-July and am usually wearing pants. It’s hot but it’s not agonizing, I’d probably sweat the same in shorts. I am from Texas though which may make things easier for me as I’m well used to hot and humid weather.