Ted,
Your problem seems related to one I'v had twice in over ten years of owning our Class C motorhome.
Your coach batteries are connected to your coach's 12 volt system with a solenoid (eletrically activated switch). These solenoids can develop a frustrating failure mode in which they are opening and closing all right whenever you operate the coach battery on/off switch on the wall (mine is by the coach entrance stairwell) ... such that the contacts of the solenoid have too much resistance between them when they're closed. What this can do is cause 13.7-13.8 volts to be read on the side of the solenoid coming from the convert and something less to be read on the other side that's connected to the coach batteries -> whatever voltage the coach batteries are resting at. The coach batteries can be kept charged partially by the converter but not all the way and whenever you check, on one side of the solenoid shows full converter voltage but on the other side you see the lower battery voltage.
In my two solenoid failures in the past, it was the other solenoid that connects the coach batteries to the engine alternator whenever the starting key was ON (the engine running). The alternator would be delivering, say, 14.2 volts to the engine battery but the coach batteries would show less voltage on them - say 12.7 volts .... as if the alternator voltage wasn't getting applied to the coach battery terminals through the solenoid contacts. What was wrong was corroded solenoid contacts such that they would still convey current, but through too much resistance due to the corrosion on the solenoid contact surfaces. The solution was a new solenoid - which I had to install two different times over the years. A good solenoid is supposed to have zero resistance across it's contacts .... a solenoid with aged contacts will have a resistance reading (in ohms) across it's contacts. Your batteries can still supply power or be charged through the solenoid's corroded contacts, but not very well.
Maybe I've misread something about your problem - but be warned - 12 volt battery system solenoids are not always doing just the simple function they're supposed to be doing.