Alternator should be charging all the batteries; provided the charge rate is high enough.
The charge system on my coach , wants to see a charge rate over 13.2 volts for over 30 seconds at the coach batteries , before it connects all the batteries together.
It will do the same on shore power, when the charge of the coach batteries from the converter/charger is over 13.2v, it connects and charges the chassis batteries.
Testing of the system is easy , volt meter in hand with the engine running, for over a couple of mins., check voltage at both battery banks . Volts should be in the 13.5 >14.2v range and the same at both sets of batteries.
Do the same on shore power, to see if your coach is set up to charge the chassis batteries when you're in a site.
99DSDP 3884, Freightliner, XC, CAT 3126B, 300 HP /ALLISON 3060
2000 Caravan toad, Remco & Blue Ox.