Propane is great for fridge -- not the battery
Propane is amazingly efficient -- we have had our fridge on propane for a month at a time -- we probably could go a month on one tank but have always refilled every two weeks, just in case.
The battery is another story -- the fridge drains it pretty quick. In our new X-139, they no longer even give us the option of powering it with a battery -- it is on 110 current or propane, period.
I should add that when you are running the fridge on propane, it still uses a little battery juice for the control panel, but not much.