When not on shore power, most of the required energy comes from propane. You won't need a lot of 12V DC power while towing, so it should run.
Some people advise against running on propane while towing due to fire hazard. The primary risk is at fuel pumps. While diesel isn't too risky, vapor from a nearby gas pump presents a greater risk.
That said, many, many people travel with refrigerator running on propane without incident.