It would help to know what type of RV you have. Do your house electric items work when on shore power? If you can even plug it in to a 15 amp wall outlet it should give limited power through your converter/charger or inverter/charger.
Your transfer switch only works to switch between 120 v sources, shore power or generator. An inverter changes 12v battery power to 120 v AC unless it is an inverter/charger. That would convert 120 v shore power or generator power to 12 v power, then when not plugged in it provides limited 120 v AC power from the batteries.
Have you taken a voltage reading at your house batteries? You have power on the chassis items, so the chassis batteries seem O.K. When testing the house batteries, it's a good idea to unbolt the cable clamps and clean the connections, they could be the problem. Next, the house batteries might be turned off at the battery isolation switch, usually near the entrance. If you have voltage at the batteries, then the next thing to check would be the battery isolation solenoid. Also near that solenoid should be a large (80 amp?) fuse or circuit breaker to protect your wiring. If that is blown, no power past that point.