As others have said, yes the generator does charge house batteries.
Since you're only going for a few days dry camping all should be fine. You'll want to crank up the genny in the morning to run the microwave, hair dryer, coffee maker, etc. anyway. While it's running the batteries will be charging. Run it for at least 1 hour in the morning.
You may want to run it in the evening too for coffee maker, crock pot, etc. Any appliance that draws a lot of electricity just crank up the genny and run it. Your batteries will charge.
When the genny is off, limit your use of power to only necessary lights, TV, etc. Don't run your high wattage appliances off the inverter. You should be able to run a TV, Satellite receiver, a couple of lights, laptop computer, etc. in the evening with no problem. (run the laptop on it's own battery, then plug it in when the generator is running and charge it).
Be sure that you eliminate unnecessary draws on the batteries. Ensure the fridge is in GAS Only mode. If it has a "climate control" switch or "moisture" switch turn it OFF (it draws DC Voltage to heat an element).
Here's a link to more detail about 12v electric for dry camping.
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