A combined inverter/charger has it's own internal transfer switch, it sends incoming AC power directly to the outlets while the inverter changes into a charger when outside AC power is present.
Does any 120 volt appliance NOT on the inverter work when the generator or shore power is available? The air conditioner, the electric side of the hot water heater, etc? If these work, the generator transfer switch is most likely OK and is sending power to the inverter like it should, and the inverter's internal transfer switch is messing up and not sending that power through to the inverter outlets.
If nothing else works, it's likely the generator / shore power transfer switch is faulty.