Good info and good description of some of the problems, so we can get down to it!
So if I read it rigth most all the inside 12VDC things like lights, fans, etc work when plugged in and the converter is sending power to the batteries and other things like lights, but they stop when the converter is not working when RV is not plugged in.
I would look seriously at the ground strap on the coach batteries for the following reasons.
When plugged, the converter is connected to the 12VDC things and the batteries. We often fall into thinking of the battery as power going in and then coming out the other side which messes with our mind as it is more like a storage area where we put the extra and "may" need to pull power out of storage. A "side room" more than a pass through?
The problem can come in when we have a good full circle (circuit?) with the converter but not the batteries! The power from the batteries can only work if it has a full circle from the ground to the batteries, to the item we want to power , through that and to ground again!
Since there has been recent battery work, that is always a suspect point but also the circle seems to be open/broken from ground to battery as the most suspect. We could look at the battery disconnect switch and relay but notif it works with the power plugged in, so I skip that .
But it is easy for a big ground cable running from a ground buss bar on the frame and hidden down behind the batteries where it may fall back and get lost!
The chassis battery is shown on left as we look at this drawing but I think if looking down from outside, this may be turned around. There should be one single battery for starting and two tied together for coach items!
Go with what you see if it looks backwards!!
Click this snip for better view.
This is the laytout I would expect, but the cable colors can get changed over the years so possibly ignore that if your cables don't match colors but do look for the locations and positive (+) and negative (-) with the negative being connected to the buss bar. Maybe hiding out of sight?
I suspect a lost or looose/corroded ground /negative cable!
But I'm also open to being wrong on my guesses so let us know??