Mircowave and hair dryer will drain your batteries so fast you will find that you are much better off running the generator for those short periods of usage unless you are willing to pay for a battery bank like those found in a Tesla electric car.
The chassis battery should not even be a concern unless you plan to be parked for weeks or months without starting the engine. If that's the case (and you setup does not charge the chassis batteries) there are stand alone and add-on chassis battery chargers.
Not only do you need a good inverter, you need to ensure that inverter has enough charging capacity to charge your batteries as quickly as possible. Charging your batteries will require that you run your generator. If you don't want to do that then you need to install a whole bunch of solar panels but since they do no good after dark you will still likely run the batteries down running the microwave at night and a hairdryer in the morning unless you plan to install a whole bunch of battery capacity (and even then.....).
