I agree on it being unusual for a generator to charge the chassis batteries.
That is why I would suggest installing a permanent on board chassis battery charger sized properly to the batteries. Just makes life a little easier when you get parked so you don't have to drag out a battery tender and extension cord to open a battery box and connect everything.
IOTA makes a compact reasonably priced charger that could be mounted in a vented compartment easily. They come in various ratings but are all the same physical dimensions. I would recommend on wiring into the switched A/C power source so it would be on with or without shore power connection so any time there is A/C power you would be charging the chassis batteries. You could put it on a switched A/C power circuit to avoid turning it on while operating the inverter.
I am assuming most people do not run their generator while driving down the road. We all know what happens when you "Assume" things though...
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