A good test is to use a MULTIMETER across the house battery terminals...
Without shore power hooked up or the MH engine running measure the DC VOLTAGE at the battery terminals. It will read 12.6-7VDC if the batteries are fully charged.
Then go start the MH engine. Measuring the DC VOLTAGE at the same battery terminals should jump to 13.6 or higher indicating they are being charged from the motor alternator system...
You can get one of these very useful multimeter from LOWES-WALMART-AMAZON or any number of auto parts stores for $5 to $20 dollars...
Looks like this google photo...
Most of us rv'ers carry one of these multimeters in our RV tool box...
Roy Ken