Yes, you can do that, but I suggest a voltage-sensing combiner between the house bus and the battery so that the house and genset batteries are only connected when being charged.
I’m using two combiners. One connects the house and engine battery when there is charging voltage present. The other connects the house 12-volt system to the generator battery. Charging voltage is 13.3 V or higher. Without charging voltage, the combiners separate the batteries so that they don’t attempt to charge each other to destruction.
The Intellipower converter/charger is much better for batteries than the charger in the genset, which one could disconnect.
My combiners are Yandina models, and I use the smallest one for the generator battery.
The only caveat is don’t mix battery chemistries. All three of my batteries are conventional flooded-cell batteries.
Rick “just replaced batteries that were 7 and 11 years old” Denney