If your battery is good, then there should be absolutely no problem running your fridge on propane for weeks. The fridge will take very little battery charge to run on propane.
I dry camp on battery and propane only. I can camp for 4 days: fridge on propane, water pump turned on all the time (it only runs when the faucets are open), lighting, charging/running netbook, cells, mp3 player with speakers, water heater on propane, propane stove/oven, etc. and STILL NOT dent my battery charge (or my propane).