Pat, truth is, folks do it both ways.
Some run powering their refrigerator
with DC. Some with GAS. Many of the larger refrigerators in Class C's and A's only have GAS and electric mode, so they are all running with the gas lite (some run with the refrig on OFF.)
Obviously you are safer running with DC. But many feel as long as you remember to turn off the propane
at fuel stops, you'll be ok.
Like Ches says ... if you know you are headed out for an extended period of dry camping
... you might try to conserve all your battery
power for the campout.
Sometimes, we even run with the refrigerator
in the OFF position! Here's how...
Usually, the first day traveling for us is the longest. Several days before leaving, I purchase a jug of orange juice, drink a glass and then freeze the rest. Also make up and freeze a two-litre of lemonade.
In the morning before we leave, I pull out the frozen juice and lemonade, put in in the food compartment, along with a couple of pre-frozen meals (breakfast casserole and maybe a pasta dish) and then we'll run like crazy all day often into the night ... with the refrigerator on OFF!
The multitude of pre-frozen articles keeps the refrigerator cold for several days! The juice doesn't thaw for atleast 36 hours ... keeping things plenty cold.
Remember, however, that we rarely, if ever, run in the extreme heat.
For years, we've headed for the mountains or Canada at the first sign of heat.
I imagine if you are planning a lengthwise run down Texas in the summer, you'd have to run the refrigerator while you travel.