The towing while running the LP is a sometimes hot topic. (yes that was supposed to be a pun!)
The issue with running the refer on propane is the open flame, while under full load of the burner, the pilot for those with a pilot, or the igniter that is 12 volt
All of the above are ignition sources. What could you ignite you ask? Fumes from the yahoo next to you at a gas station that dumps $2.00 of gas on the ground, A VERY big pocket of Methane gas, or other flammable gas.
I for one am more worried about the 50 gallons of gasoline I carry in a frame mounted tank. The propane will vent and will require a large concentration to cause an explosion.
I am also more concerned regarding the possibility of food poisoning due to the refer getting too warm. I run mine on gas all the time I travel, many of the Airstreamers I know do.