This doesn't answer your question, but I spend a significant amount of time on weekends out training my dogs with a group so I've spent a lot of time looking into questions like this.
I haven't thought of doing that specifically (small AC unit plugged into bed outlet). Primarily I combat the heat by keeping the vented windows of the canopy open at all times, prioritizing parking in the shade (or using a shade cloth when I can't), using portable battery fans, and tons of fresh water. I keep a SensorPush thermometer in the bed so I can monitor via my phone as well. The temperature noticeably drops while driving I've found. And it was the main factor for me to get a White truck instead of Black.
I've seen some people make a swamp cooler out of an old cooler by cutting holes for fan/exhaust and filling it with ice. With a portable battery fan then you could do this without the need for the truck to be running.