As Styrene is a potential carcinogen, me, I’d likely not want steaming water vapours regularly flowing through my home from a styrofoam vessel housing a hot water bath.
Just about any cooler will work to use as an insulated vessel. Not a bad idea to contact the manufacturer of the cooler in advance to query if the plastics they’ve used will stand up to hotter temperatures in advance. (they likely will, as they often use the same materials for insulated drink vessels).
If the lid of the cooler is hollow after you cut the hole (using a hole cutting bit) you can try using some spray foam to fill the gap and provide some additional strength - just be careful as you’re applying as it expands as you don’t want it misshaping the lit. Then trim it up and seal the edge of your hole with duct tape. You can then use additional duct tape or electrical tape to get a really good seal for the hole. This will prevent any water vapours escaping and causing havoc with the cooling of your APC.
When you look at your APC, the exhaust vent for the heating area juts out - it’s the lower lip of that, that you want to sit on the top of your cooler lid.