It sounds to me like Steve is asking about the amount of time that the fridge should run off batteries, not if it runs. Otherwise I would not have expected the reference about when engine is running. The inverter would still be powering the fridge, right? The difference being that engine alternator can keep batteries charged, and hence powering fridge.
I may be interpreting his question completely wrong, but it sounds like his issue is that batteries are not powering the fridge for a couple of hours, which may suggest bad batteries.
Perhaps he can clarify what is happening. I don't own one of these so can't help with specifics.
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