Hello Big Tim
I have read your postings rather carefully, I think it may help to clarify the terms I use to avoid confusion latter on,
EHU = Mains
PSU/charger = 12/13.8V Power Supply Unit
Switch Panel = distibution panel for each 12V circuit
Let me first say that the battery condition meters I have usually seen in caravans, are usually pretty crude moving iron meter which actually uses a measurable current to make it indicate. That may be the small current you found when every thing else is turned off. The other thing about these meters is they are not especially accurate, which you have confirmed by your comparative readings with the DMM. So for accuracy I suggest you ignore the caravans meter.
From the results you have been getting by measuring the voltages at the battery, it seems to be holding up quite well, so I do not think the battery is at fault.
Now we come to the charging circuit. You tell us that you have discovered the 12v power supply unit (PSU/Charger) is rated at 13.8V 20A. This is a goods size and should be capable of running the caravan without a battery, with the exception of caravan movers.
If I have understood the situation correctly, you tell us that if you completely remove thy battery from the caravan, and connect the caravan to the EHU, that despite juggling with the switches you cannot get any of the 12V appliances to work. That is pretty conclusive and lead one to suspect the PSU is not producing any But you also tell us that if you measure the Voltage at the battery clamps(no battery connected) that you see 9.5V. put the battery back in the caravan and leave it on the EHU, the battery Voltage does slowly rise to wards 13.8V.
This suggests to that the fuses are intact otherwise there would be no transfer of charge at all.
This all suggests to me there is a fault with the PSU, it is partly working, capable of only providing trickle charge, rather than the full 20A.
based on the above I strongly suspect the PSU is at fault here.