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I have all of the equipment to do what you've said you want to do. I have a 2kw inverter/charger with a support/generate function. It works like this... You program the inverter with the amount of generator/shore power available down to 10 amps. This works perfectly with my 1200 watt Honda. The generator is running and powering everything through the inverter bypass. If you say press start on the microwave and the house requires more than 10 amps the inverter will draw from the battery to support the load and supplement the generator. When the load drops the surplus is then used to charge the battery. This inverter was not an inexpensive piece of equipment like the inverters you cite. Mine is meant for boats and RVs with limited shore power.
In your case I would find a large 12V alternator and engine to charge the batteries as needed. Add some solar if you like to help keep the fuel consumption down. Under normal grid conditions you can use the solar with a zero-export inverter to power household loads.
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