Living in Phoenix, Arizona where the temps hits over 100F for something like half the year, I've given some thought on how to keep my hamster cool. Some ideas that come to mind
1. The biggest problem to temperature control will be when you stop the car and shut off the AC. I don't know what your driving style is, but maybe you could take short breaks where you keep the vehicle, and thus AC, running
2. Other threads have said to provide a cucumber for water and food
3. Depending on what your travel cage looks like, perhaps you could strap a water bottle filled with cold water to the outside of the cage with the nozzle inside the cage. Or of you can't do that, maybe you could zip tie it inside the cage. You don't want it loose to potentially fall on the hamster. Potential problems: it could drip, or if your hamster is a chewer could chew through the zip ties. Maybe bring extra bedding to change out if it drips
4. There are reusable ice packs that you freeze and put in in coolers. Maybe you could get a couple of those and strap one either on the top or side of the outside of the cage to provide an area that's cooler. Then provide a lot of bedding so he can burrow away from it if it's too cold.
5. I have a Syrian male. When summer hit (we started breaking 80sF back in March or April) I started noticing that his testicles get really huge looking when he's warm. It's like a built in thermometer. He's never exhibited behaviors of hyperthermia that I've read about, like looking listless or flattening himself out along a cool surface, and he has lots of different cooling materials in his cage (marble slab, rocks) that he does not seek out to cool down, but it makes me feel better to put him in a play pen with a tile surface to cool him down when I see his testicles looking larger than normal. If you have a syrian male maybe you could use that as a gauge if he's starting to be too warm.
6. Complicated option: You could put the whole travel cage into a cooler, leave the top off for air flow. You could have a couple of the ice packs that you put in to the fridge rather than the freezer so it's not too cold, put them on the outsides of the cage, in the cooler. Bring back up ice packs in a separate cooler (with lid on). Have a thermometer where the sensors are at the tip of a wire, have that in the hamster cooler, stick the readout of the thermometer on your dashboard. Now you have a monitored environment.
Quick online search of thermometers with wire sensors for a few examples:
https://www.acurite.com/digital-ther...ed-sensor.html
Note: I haven't had any need to travel with my hamster so I haven't actually tried any of these out, so hopefully there's at least one idea here that you find useful.