Remember, when you see people running fridges on 12v here they are almost universally compressor fridges which are designed to run on 12v fairly efficiently. A 3-way fridge draws a lot of current if you run it on 12v instead of propane as it the 12v is running a little heater in place of the propane flame. I imagine if you have a modern 3-way fridge it will have some form of battery protection circuitry, but I recall my old 3-way had nothing of the sort, and a small heater element like that will just keep draining and draining the battery until it is completely dead and fit only for the trash can.
Steve.