The fridge uses 12v when in 120vac mode too, so it must be getting 12v power. Do other LP appliances work when the fridge does not (stove, water heater)?
Maybe the 12v power is low without the converter running (shore power)? With converter running, the voltage should be around 13.3-13.6. If you are getting lower voltage when on battery alone, maybe its not enough to open the gas solenoid valve on the fridge. It's supposed to work as low as 10.5v, but sometimes they lie!
I have a satellite dome that is supposed to work on 11.5v, but in fact it needs 13+ to function. The dome manufacturer denied it, but "advised" that their optional booster would solve my problem. And it did. But they refused to pay for it.