My generator does not charge the chassis batteries, hence the need for the aux battery charger, which Truk and I installed and it has worked like a charm.
Once you have yours installed, that will be your ticket to charging the chassis batteries while running the generator.
Just something for you to think about, the charge rate setting on your inverter/charger(s) will impact the time required for your generator to run to fully charge your batteries. Of course, the beginning state of charge will also matter.