I have always ran mine on gas/propane when traveling..... and never had any problems with the fridge staying cold... one thing that helps keep the fridge cool and the odors of a warm fridge at bay, is to install a fan inside the fridge near the bottom so it blows the air upwards....
I have a small 3x3x1.25inch squirrel cage fan that i found on ebay, that Ive mounted with double sided tape between the bottom drawers that blows up towards the cooling fins, that runs on a timer (12v) that activates anytime the fridge is turned on.... it blows for 1 minute every 20minutes, which keeps the interior at an even, constant temp all the time... it makes a HUGE difference in keeping the temps down when the outside temps get near 90 and above.... and no odors ever.... I used this same fan/timer set up in my old trailer for 4 years, and when we got this trailer 2 years ago, i moved it to this one....
the generator can run while you are traveling, but there really is no need unless you have dead batteries and need to run charger/inverter while moving....