Just got a new Coachmen Clipper, and the fridge has Auto mode, which uses shore power as the first option, and auto switches to propane when TT isn't plugged in. It does not run off battery at all. The only things working off the battery are lights, radio, and CO/propane safety detectors, and break-away switch. Fridge, AC, microwave, only work w/ shore power.
Haven't gone camping yet but will be taking others advice, pre-chill the fridge overnight ahead of time, put frozen/chilled foods in for the drive.
Great tip on lowering the temp while traveling on propane to preserve, will probably try that as well.