Obviously something is wired wrongly, but you know that already.
The 30A breaker from the inverter--does it really and truly go to the inverter (or its associated transfer switch)?
I would have guessed that the two-way fridge would be one thing that should not be run from inverter power typically. A residential compressor fridge is, of course, a different matter altogether. I would probably tackle this problem by removing all power from the system, turning off all the breakers, and using a meter or signal tracer figure out exactly what is connected to each of the breaker outputs. It might be a bit of a forensic search.