I'm in the same heat dome as you so I know your concerns. The bucket is fine. You say you're scared it will be too hot for them - too hot in the car? In the dorm, before you can get the room acclimated and cooled down? In the tank, after it sits in the car overnight and retains heat even after you add water?
You have AC in your car, right? Whatever is comfortable for you will be comfortable for them (unless you like super-low meat locker temps). Just don't point the vents directly at them.
If you're walking into a hot dorm room, switch on the AC immediately. A room that small will be comfortable in 10-15 minutes, which is fine. Just get everyone out of the car and into the room immediately, before the luggage and boxes.
Getting the tank set up is the priority, so adding water will cool down the hot glass and substrate. Watch your thermometer and once it's 2/3 full, adjust with cooler or warmer water a little at a time. Then add your guys. I've seen your prior posts, sounds like you have a pretty good idea of how long they can hang out in their temporary homes until the tank is ready.
During this heat wave I took a 4 day trip and came home to find that my AC had quit. I walked into a house that was 99º. The tank temps were same as the room. While someone else handled the AC trouble-shooting and repair, my first priority was the livestock. At first I didn't see anyone and thought they'd all perished.
The bettas were the first to emerge, to wearily greet me. They'd been in the darkest, safest-feeling places (the betta log), battened down in emergency mode. The shrimp were next, coming out from the depths of lotus pods and moving slowly. The snails were last. They too had gone into emergency mode, burying themselves in substrate and didn't surface until a full 48 hours after the tanks were back to normal.
They way I did that was to scoop out two cups of water at a time, slowly replacing it with water from the fridge. Slowly the temps dropped. I resisted the urge to bring it down fast: they'd made it this far, a fast drop could've shocked them to death.
To my surprise, everyone survived. The bettas lost color from the stress; ten days later they were back to normal. All of them paced the front of their tanks following my every move for a week; I'm guessing they were traumatized at being abandoned almost to death.
I don't know how long the house was that hot. I'm guessing, based on surface heat of furnishings and also cans on kitchen shelves that exploded (yes. don't ask.) it was around 12 hours. I'm certain that had I returned 24 hours after it quit, most of them would've been dead.
My point: although I would never, ever recommend letting things go beyond their comfort zone, the hardiness of these guys surprised me. With all the care and concern you're showing, I'm sure you'll transport your guys unharmed. Good luck tomorrow.
