Hi:wpski... Walt, we have never travelled with the fridge on. Just make sure to turn it on as soon as you get near a "Currant Bush", or use the propane in a Wall o Mart parking lot.
I am sure that it is safe to tow using the propane and it should be wind proof but in the event the flame dies the propane shuts off auto pronto!!! You don't want it on while filling the tug... and lots of places won't let TT's or RV's in with it on!!!
Our fridge stays good and cold with the door closed and the beer is ready to drink while getting site set. I always take cans and use a hugger to prevent frost bite.
escape artist N.S. of Lake Erie