Just rolled over 9000 miles on X VIN 265. AC works great on Super Chargers or I would be dead. Charged at Yuma AZ last week, Hot Batteries ( Ran non stop, high speed, from full range charge to less than 10 miles range remaining, mid day, parked full sun on hot asphalt)
The car cooled just fine. Even on range mode.
If your AC is not working while charging try a system reboot and if that doesn't clear it call service.
In over three years and 75,000 miles of Tesla driving, Model S and Model X, I have charged at the hottest locations in all of Tesla land many times. Barstow CA before the solar shade, Quartzite AZ, Gila Bend AZ, Yuma AZ, El Centro CA and Primm NV. I don't think Super Chargers come any hotter in the summer than those locations.
In all that time and all those charges I have experienced limited interior cooling only once. Mid day in El Centro CA with the car temp gage reading 124 degrees outside, as I was charging hot batteries, parked over smoking hot asphalt, in full mid day sun. After about 10 minutes charging the car displayed a warning that interior cooling would be limited. I was not surprised to get the warning. It was incredibly hot that day.
So you should have no problem getting full cooling while Super Charging. After hundreds of S and X Super Charges on road trips I have only one time had cooling limited.