Yes, the alternator should charge both battery banks.
Confirm with a digital voltmeter:
With engine off, shore power disconnected, generator off, check voltage at the chassis battery and then at the house battery. 12.2 is a 50% discharged battery. 12.7 is a fully charged battery. If above 12.7, it is just surface charge. Turn on a couple of lights for a few minutes (or other load) to dissipate the surface charge. You want to start this test with voltage at or below 12.7.
Start the engine and put it at high idle (around 1100 RPM). Recheck voltage at the batteries. Voltage how should be in the 13.5- 14.2 VDC.
There are two different systems for allowing the charging of batteries by the alternator, yet separating them during discharging. Diode-based isolator or solenoid-based isolator.
Let us know what your voltage readings are and if you know it, which type of isolator you have and we can talk you through troubleshooting it.
