Just keep him in the fridge and do salt baths until 3 days after the fungus is gone.
Floating isn't always a sign of illness, only if they can't get to the bottom when they want to and are struggling.
Have you got a water test kit? If not it will be a good idea to invest in one, and if you have one what are the parameters of the tank water?
The heat would have stressed him a lot been that high which can lead to illness that quickly. How long was the water that hot for?
He wont be active in the fridge, but one the fungus is gone I think he should be fine, just make sure to keep his tank cool and keep an eye on the water parameters. It's good that he is still eating.
And I haven't dealt with a dead axie before, but mine have often just sat really still in weird positions and made me think they were dead, I poked them and they swim away after biting me, or in some cases stay still so I lift them up and then jump and swim down to the bottom of the tank, so I imagine if they were dead you could tell since they wouldn't get upset at been touched/picked up.
Good Luck with him! I hope your kids and your axie are feeling better soon