Okay, I see some confusion creeping in here as we are not really speaking of an inverter! An inverter USES 12VDC to "make" 110AC!
BUT what you have is actually an inverter charger! And it does charge the batteries!
But an inverter would not!
Right place but wrong name?
I would look at this as most likely a corroded/dirty cable until proved different. Far cheaper option to fix, too!
I might start with doing some basic checks for corroded cables to find which direction to move the chase.
You likely have spotted an area next to the batteries like this?
Click to see better!
Spot detail CB I marked ? It is likely behind a panel but that is where lots of things need to happen and we need to look at that detail CB in the next picture.
This is the place where the chassis tuff is on the left side and coach on the right. In between there is a battery disconnect relay for each chassis and coach that I marked as orange and blue.
Between those is the BIM ( battery isolation manager). That BIM is what controls when the two are connected or disconnected and that can be a biggie to know!
I favor going here first as it is a convenient place to check both battery groups and how they are connected or disconnected! If the cables are not getting to this spot well, the charge is not going to get from the 12VDC charge coming off the charger to the batteries!
I might favor checking the voltage of the coach batteries and the chassis batteries when the power cord is not plugged in and the batteries have not been charging for an hour or so to let them become stable for best readings.
Best if you read directly on the battery posts and not on the clamps or other parts which may not be connected well!
Those readings should be less than 12.8 but note exactly what you get on coach and chassis!
Then move to this set of of connections and check that you get near the same readings here for both! If they are more than half volt different that the battery readings, there is likely to be a cable problem to chase!
But if they look close to the first readings, then plug in the power cord to let the charger start charging or maintaining the batteries. That higher reading should show up higher on the coach side to charge those batteries.
Whether the chassis batteries side jumps higher will depend on how the BIM feels about the voltage there.
If the chassis battery is lower than 12, that is where the lack of charging is the problem!
You have two disconnect relays, one for each , coach and chassis.
When those are closed as they should be when using the RV, is the voltage the same on each side of the relays?
When the disconnect switches are on, the voltage on each of the relays should just pass through these relays. If not, the relay may have bad contacts, etc.
Basic point is to check the battery voltage from the chassis batty reads the same here as at the battery. If it get s here from the battery and we see charge voltage here, it should also get back the other way to the battery!
So this is a place where you can do a fair amount of checking to limit the number of other places to look! If you don't get the same voltage here as at the batteries, there may be a big mega fuse inline that needs checking!
Good luck on the chase!