Your engine's alternator is the device that generates the charging current (about 14+ volts) which is sent to the chassis batteries and house batteries. Our coach has a BIRD/Big Boy system to allow the charging current to move between the chassis and house battery banks. Your coach may have an Echo Charger instead of a BIRD/Big Boy system.
When plugged into shore power or when running the generator (both about 120 volts), the "inverter/charger" generates a 13+ volt charging current which is sent to the house batteries and the chassis batteries. Again, our coach has a BIRD/Big Boy system to allow the charging current to move between the house and chassis battery banks. Your coach may have an Echo Charger instead of a BIRD/Big Boy system.
If your coach does not have an inverter/charger (or some other charger-like device), then your coach cannot charge its battery banks while plugged into shore power or when running the generator.
In the past, our inverter/charger stopped working and had to be replaced. Before getting it replaced, we used two portable chargers (plugged into 120-volt receptacles) to charge the two battery banks (house and chassis).