First to clear naming confusion as many do not use the correct names. An inverter uses AC to "make " DC. It doesn't charge anything!
A converter does the opposite and uses AC to make DC!
However there is a combined unit that is an inverter charger that does both and it is often gets called "inverter", even though it is not!
There is a point to know about your RV and what exactly you have as there is an early build as well as later year build. Early has a 1 as the 7th digit of the serial number! That is the SERIAL number on the tag next to the driver's left side. NOT the VIN that is used for license, etc.
A 2 there means later year build and may have some small points different, so it can be important to know which you have!
But if you have the inverter, you will not have an inverter charger! So expect to find a converter located on the floor behind the load center where the breakers and fuses are found!
Some tricks to sort the problem may help. If you have a meter to test voltage, put it on the coach batteries and watch what happens when you plug or unplug the RV. If the battery voltage jumps when you plug in, the converter voltage is what you see there and it is working. Maybe the converter is sendingthe charge to the batteries but they just won' hold it?
But it helps to know how it gets there as this drawing shows and there are things to trick us!
Some guessing on early or late build but I think your load center will look like this. with AC breakers on left and DC fuses on right, we can look at the left side for how the converter gets power.
Power in on the cord from campground plug or generastor gopes to main breaker and I think it feeds the whole load center but it may only go to the converter main. Not shown on drawings, so a bit of guess there? One or both those breakers should let AC go to the converter that is behind the load center. That converter makes the DC and feeds it out to both the batteries to charge then and also to the fuses for RV items.
A tricky trap to watch is where the battery disconnect switch is located as a trap!
That disconnect switch cuts off the batteries from charging but not from discharging!
It cuts the batteries off from charge but not from the fuses that can run them down!
Check/verify anything you read on the internet, so test this info before trusting?
With the RV plugged in and the converter working, turn the disconnect off and see if the lights still work? They should as they are fed by the converter. Now disconnect the cord and I think you will fidn the lights stay on as they are working off the battyeries!
Ugly trick that can kill your first set of batteries if you run them totally dead while stored with the RV plugged but the disconnect off!
But to sort the converter charge question, some suggestions. Maybe a first easy one if it is quite where you have the RV, is to listen for it to hum! With the RV plugged and all disocnenect and breakers on, you can often hear the converter hum when near the load center! Converter is a transformer if you know what they sound like? A 60 cycle hum? No hum, chase why power is not getting from cord to breakers and converter, bu tif there is humn, it should be getting through the disconnect to the batteries and there can be corroded connections, etc, on that path.
One point to be aware of is the voltage you mention as that is pretty low for a set of fully charged batteries as they should be at least 12.7 when not charging and jump higher when they get charged as you will see the higher voltage going in almost immediatly, even if the battery is not fully charged itself. Like pouring water into a jug? You should see the water going in even if the jug is empty.
