I doubt the inverter needs to be on to charge from the alternator. The charger function in inverter/chargers is totally separate from the inverter portion of the unit and the charger works anytime you are plugged into shore power or generator.
The alternator is a "charger" on it's own is is connected directly to the batteries.
I don't have any experience with Sprinter chassis motorhomes but I think that there is a fuse under the driver's chair that can fail and house batteries will not charge from the alternator. I have also heard sometimes it's not the fuse but a BIM solenoid.
I hope I'm correct with this, but wanted to mention it just in case it helps.
This video below is for a non-Winnebago coach so the control panels shown are totally different, etc. But the fuse under the seat should be similar.