Sooo... Running on LP is mostly legal except for certain points which may make it legal, but kind of a PITA.
With our big trailer I run the fridge
on 12V while traveling and switch to shorepower once we set up. Prior to that (older trailer didn't have a battery) I did the frozen bottle thing. Worked fine for up to 6 hours or so as long as a pre-cooled the fridge
at home and loaded it full of already cold stuff. Depending on the timing of your menu - some other items can be frozen too. My wife and I used to freeze Cornish Game Hens and then take them backpacking in just a small (flexable, not very good) cooler. By the time they thawed out we'd be ready to grill them.