Good job of getting the right info to us to do some really firm looking!
Good questions usually can get better answers.
So several points which "might" give problems and which to look at first may depend on which you can find and get to easier, so I'll suggest and you decide?
I might first want to be sure the batteries are set up and in shape to take and hold the charge. Age, condition but often more trouble is loose or corroded cables, so I would look at those both at the coach batteries and also the big cable that goes to ground, often on the frame closeby. Bad ground kills things just like bad hot wire!
Then I did a snip of the drawing for your RV and marked up the path I see. Let me mention some things to help if you are not familiar with wiring that a small number like gauge 1 is a big finger size wire and larger numbers like 8 is smaller wire like 1/8- 1/4 inch thick and this drawing gives sizes and colors to help sort which wire might go where!
Top solenoid in this drawing should be what you hear when pressing the aux button as it is kind of a temporary connection between start and coach battery string (like a jumper cable?) and I don't think it will be involved with getting the coach batteries charged from the converter.
But the one below sounds like it is working as it disconnects most of the RV from the coach batteries during storage, so if it was not passing current through you likely would get NO inside lights, water pump and all those 12vdc things. So the connection from coach battery (red dots) is connecting okay to the number 8 black wire on the right (black dots) assuming the wires okay, not corroded, brooken, etc. and we can trace it to right on the drawing to a 40 amp circuit breaker and that's up under the dash and likely not wanting to be found! So I might expect to move on to find the connection where option 2 in the left lower corner tells us it is in the front electrical box or if hard to find that, look for the isolated stud near the converter. I might expect a big bolt looking thing with number 8 black wire meeting another wire which likely comes from the converter!
This is where things get down to working or not, so look at the converter to make sure there are no switches, GFCI breakers or such and there should be at least 12-14 VDC coming out of the converter to this stud to go back down to the coach battery.
I would want to take the black wire off, see coach battery on the end of it and also battery coming from the converter to the stud.
If this stud is easy enough to find and work on, checking here might be a good place to start to find out if the converter is not passing voltage or is it the path to the batteries not working right.
Kind of like the hose in the yard and you want water going into a bucket? The faucet (converter) not working or is there a kink in the hose between there and the bucket (batteries)!!
Try clicking the picture to get better to read ,but if comes from this drawing: http://www.winnebagoind.com/diagram/1998/126757.pdf
See how that goes?