As of this reply, 110 folks have viewed your posting but none have replied. Your issue appears to be whether your house battery charger also charges the engine’s battery and, visa versa, whether the vehicle’s alternator charges the house batteries. Since you want to have dissimilar batteries (Li-Ion for the house and lead-acid for the engine) that’s a damn good question.
Since no one has offered a suggestion, let me take a stab - test the house batteries at rest — nothing charging them. Start the vehicle and test again. Any rise in battery voltage would indicate charging taking place. That is, the engine’s alternator is charging the house batteries. This is common, to help keep these batteries charged while traveling.
Now, with the engine off, measure the engine battery’s voltage. Do not have shore power plugged in. Then hook up shore power (120 volt) and go test the engine battery. Again, any rise in voltage would indicate charging, this time meaning the house charger is charging not only the house batteries but the engine battery. This is less common.
With this knowledge we can better determine how to charge your dissimilar batteries.