Leave it on
With the exception of certain tunnels and when in proximity to a gasoline pump, my refrigerator will always stay on from 24 hours prior to departure until we unload the refrigerator when arriving home. With the automatic switchover, that says that it will be on propane except when we are hooked to shore power.
Since I drive a diesel, I don't turn the refrigerator off when refueling at diesel-only islands. Since I usually use truck stop pumps, this is most of the time.
I also have a refrigerator thermometer and I have found that the interior temperature is out of the safe range after only a couple of hours towing in hot weather. In Texas, that is most of the time. Since we have an 8 cu ft refrigerator and seldom travel for over two weeks at a time, the refrigerator is never full enough to have a large thermal mass. The probability of food poinoning is infinitely higher than a refrigerator fire.
I find it quite odd that a lot of folks will get all shook up over towing with propane on when they are not in the trailer and then happily sleep overnight in the trailer with the refrigerator on gas.
John W. Irwin
2005 Classic 28 "Sabre-Dog III"
2013 Silverado 2500HD Duramax/Allison LTZ
WBCCI #9632, TAC TX-10