How long did you run the generator? There are delay circuits that might not have allowed time for the inverter to recognize it could connect directly and shut down. These delays could be in the connection between house and chassis systems, the inverter, etc. Also, it takes longer than you'd think to charge the batteries back to full strength.
Also, you reported a voltage of 12.4 volts, which is less than 80% charged. If the charger was putting power into the batteries while generator was running, you should have read voltages of 13.4v or more.
Batteries also take time to absorb a charge and can read high if they have just been charged and no load is on them.