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Are you getting good AC power to all the rest of the systems on shore power? IF so, I suspect that you may have one side of the transfer switch isn't working. Is this a 50 amp cord (4 prong) here? Any adapter being used? If so, you lose 1/2 of you AC service. Check that and get back to me and let me know if we need to dig deeper into this. All exchanges of Q&A are included in our original fee.
But maybe not using an adapter. When you are using an adapter like that you lose half your AC power. That is why you aren't getting the fridge to work on AC. You can do some re-wiring in the main load center and change that, but better if you don't use adapters at all. You sure can't run the A/C on that and not good to even use the microwave either when using just a regular 110 outlet. That is way too much amperage draw for that outlet. So, long story short, there is nothing wrong with anything. You just have half the power shut off when you do that. Hope that helps here. Please get back to me if you need more input on this from me.
Just checking back in to see if you were able to resolve this or do you still need additional input from me here?