If you build this circuit,you realise that you must have a manually operated switch to restart the inverter after a low battery shut down?
Two obvious methods:- one is to feed the battery via normally open relay contacts to the circuit. connect the relay coil downstream of the contacts, put a driving transistor between the coil and earth. Or put a NPN transistor in series with the 12V feed collector to battery, emitter to circuit. Connect the base via a current limiting resistor (1K?) to the collector of the driving transistor, emitter to earth. Now in either case if there is no current going into the driving transistor, then the battery supply is cut off. You need to put a push button switch across the relay contacts or collector/base of the series transistor, to get a supply voltage to get some current into the base of the driving transistor.
Frank