Absolutely use LP mode whenever AC is not available unless you have a very robust battery and charging setup.
As you found, fridge draws enough juice to kill a normal single 12V battery pretty quick.
There is no issue running it on LP while driving (contrary to some who may reply to this thread). Be aware, certain conditions, speed, wind etc can extinguish the LP flame though and you may think the fridge is on LP and it switched to DC without you knowing.
Some campers are better than others about keeping the flame lit. Our old AF is good at highway speeds up to around 75mph if there's not much wind. If it gets very windy, it goes out.