It sounds like most things have been covered. It is always hard to guess how a coach is wired and what runs through what. You may have two circuit breakers on your inverter charger that you may have overlooked.
Have you tried these items with the generator to see if they work off of electric?
If they work with the generator, then you may have a problem with the transfer switch cutting off one leg of the shore power. That is the relay that clicks when you hook up to shore power.
Have you physically turned off and switched back on each circuit breaker on the shore pedestal as well as all of the inside circuit breakers in your main panel?
Neither your furnace or your water heater would work on LP if you lost your 12 volts so I would rule that out.
Is your refrigerator working on AC or has it also switched to LP?
With my coach, the Heat pump is on the roof and it supplies cold and warm air but it ONLY works on 120 volts AC, not LP, and uses 12 volts to the thermostat. My furnace ONLY works on LP and also uses 12 volts for ignition and the thermostat.