The inverter will usually have an inline RELAY, which senses 120v power from Shore or the Generator, then simply 'passes thru' this power to the breakers/circuits that are normally powered by the Inverter.
It initially sounds like this relay is not switching over, or not 'sensing' the 120v power from the Shore or Gen - which makes me believe that either the relay is not acting correctly, OR your 'INVERTER/Charger' breaker is tripped(giving no ability for 120v power to/thru the Charger and Inverter).
Do you have a Magnum ME2012 inverter and similar ME-RC remote panel?
Is your House Batteries holding a charge, or being CHARGED when you are on shore power or the generator? The levels should show about 13.5 or higher if under the Charger's output while on shore or generator.
'14 Palazzo 33.3 bunkhouse 34' diesel
KingTailgater2 Dish HD, 100w/5a SOLAR, BlueOx
100,000+mi since '14 - US, Alaska, Canada.... formerly Georgia, then NorthCarolina, now Alabama : )