If the 120v equipment that runs through the inverter is operating with generator and shore power, then the breakers and wiring from the inverter are probably okay. 120v equipment that runs through the inverter includes all the 120v receptacles - TVs, oven, etc. The 120v power for the air conditioning, for example, do get their power through the inverter. But you're not getting only 12v from the house batteries?
I hope the inverting portion or the 12v pass-through portion of your inverter from the house batteries is not the problem. There are likely more things to check before giving up and getting a new inverter. Mr. Buker, being a retired electrical engineer, will be able to help more when he gets a thorough and more defined description of the symptoms.
Crap - I had three phone calls while typing this message. I hope it makes some sense.