Why would a bad breaker in your inverter keep your fridge from working on propane?
I would think a bad breaker would stop providing 110V to the fridge, but could still be providing the 12V it needs to run on propane.
But, you should have this problem on shore power or generator.
And, you can easily test by testing for 110V at the refrigerator plug, and by flipping the breaker to ON.
On shore power or generator power, with the fridge breaker ON, you should have 110V at the power socket where the fridge plugs in.
On my rig you can access this socket behind the outside access hatch.