You have answers you need (just pre cool, or use ice for that short trip) but im curious now...
Are you talking about running it off the battery to heat the element, or off the battery to run your circuit board while you run it off propane?
Older fridges do not need power to run off propane. Not sure what you have.
Newer ones need power to run propane which is regulated by the circuit board powered by DC, and yes, this circuit board should run off the TV's power through the trailer plug, it does not need much power.
Running a DC heater element off the TV is not a good idea, if it were able to pull enough amps through the long wire run from alternator back to trailer fridge, then it would either run your TV battery down or do some hefty damage to the alternator making it run more and run hot. I once burnt out a 130 amp alternator trying to run a fridge like this on a class-c. I learnd much since then.
* just in case someone reading is a very newbie... propane fridges, or 2-way or 3-way fridges heat the gases to make the fridge cool. sounds backwards, but is how it works, so a DC heating element makes the fridge cool, or when on propane, the hot flame makes it cool.
The 3rd way is a heating element that is AC household current, which if you run a generator this will work fine while driving. The DC and AC heating element probably are the same in one and run off either.
Driving with propane lines open is not legal always, but many have done this since the beginning of RVing. It has been discussed to death extensively with everyone's opinions and discussed more after that.
It is possible to run the fridge on propane and drive. Dont get caught. Dont tell us you did this.
