You will get many miles from a voltmeter/multimeter as a MH owner.
Switch on or off:
My reply addressed the Invert switch on the inverter/charger itself.
Even with the house battery switch turned off inside the rig when connected to shore power, the battery charger should still charge the house batteries and maybe hum.
That charger will not charge the Chassis Battery. The engine alternator takes care of that. You can install an optional Echo charger which will help transfer some house battery voltage to the chassis battery.
12.6 dcv is regarded as a "full" 12v battery. Red probe on positive terminal, black probe on the negative battery terminal.
12.1 dcv is half empty battery and time to recharge.
When changing the batteries it is possible to blow one of the two 300A Class T inline fuses. I have one in the battery compartment and one near the inverter. Google that fuse for a picture of it. This could prevent battery charging. Check them with your new meter.
Start the Generator and see if it runs the Charging for the house batteries. Should be about 13.6 dcv measured, while the genset is running, at the house battery terminals or on your dash house battery gauge.
If that is the case, your charger is working but not with shore power for some reason.
The volt meter need not be expensive. I am still using the freebies from Harbor Freight.