Using the 12 volt mode for the fridge uses a lot of battery power, anywhere from 10 to more than 20 amps depending on the size. Even if you have sufficient wiring from the tow vehicle, it puts a strain on the tow vehicle's electrical system (Older bulb type headlights use about 10 amps of power). Also an automatic disconnect should be installed to prevent draining the tow vehicle's battery when the engine isn't running.
If you must run the fridge when traveling, the LP mode is preferred. Some have difficulty keeping the heater lighted due to air turbulence and have added baffles to help out. Others always turn the fridge off when refueling due to the possibility of igniting fueling vapors.
The easiest and safest is to add some thermal mass (ice packs) inside the fridge and this will keep it cool for many hours. Keeping the fridge filled of cans of cold beer will have the same effect.
Our Aliner has a three way fridge so we run it on 12 volt while on the road and switch to gas or electric once camp is set up. been doing this since 2005 and fridge still works great.
Lee & Jane Ford died once to often. Replaced with 2019 GMC Canyon Aliner, soon to be gone.
If it is an absorption type fridge in A-liner that runs on propane like the one in the truck camper, then a fridge is a fridge is a fridge. I see no reason not to run the fridge in transit, though I never do. Mine will stay plenty cold for at least 5 hours in transit and I rarely travel over 4 hours at a time, so....
"Sometimes I just sit and think. Sometimes I just sit." "Great minds like a think."