If you mean the charger reads 13.5V AND the battery reads 13.5V (at the battery posts) then disconnecting the battery and it drops to 11.5V then you likely have a bad battery. Have the batteries load tested.
Shore power and gen both supply AC power to the charger to maintain the batteries so expect the same for both. The gen may use some DC power and the batteries can drain w/o charging.
BTW Inverters convert 12V DC to 120V AC. They do not charge batteries. If you have a charger/inverter unit then the charger portion charges the batteries.