Considering your symptoms that everything works properly when you are on Shore power or on your Genset, then your systems are operating OK.
I suggest that you use a multimeter and check the voltage across your COACH (House) batteries and make sure they are fully charged.
You should read around 13.5-VDC if the battery is fully charged. less if it is not. I suspect that your inside equipment is operating on your Converter while you have power and when you don't have AC power, then the battery is not handling the load, which is why I suspect your battery as the prime culprit.
A long shot is that you have a loose connection on your COACh battery terminals.