A caveat: I don't bring a toad with me when I go camping, so this is not personal experience.
I would use the tow dolly long before towing four down with the engine on. Idling an engine for a long period of time is not particularly good for the engine, wastes gas (albeit not a lot of gas), and seems to me to be almost asking for trouble with e.g. the transmission bouncing or getting bumped out of neutral or if the engine stalls or runs out of gas. Mileage while towing will accumulate on the minivan's odometer, too.
Overheating should not be a problem. Modern engines don't produce too much heat when idling, and the electric radiator fans are quite effective at dissipating whatever heat is produced if natural airflow does not do that.