I thought the rear 12V sockets and reading lights worked off grid, how does my coolbox work when driving and wouldn't it drain the leisure battery if I left it plugged in with the engine off. The 13.9 reading is on hook up and using a volt meter that plugs into the 12V socket. Looking at my solar panel app on my phone the battery is showing 12.9 without hookup. I'm starting to wonder whether something hasn't been reconnected when the heater was fitted.
The 12v sockets and ceiling lights are powered from the Leisure Battery as is your fridge if plugged into a 12v socket in the habitation area.
When plugged into the electrical mains a Battery charger charges the 12v Leisure Battery hence the voltage reading of 13.9 volt, and also the Mains socket becomes live.
When the engine is running the alternator will charge both the engine battery and leisure battery and if you measure the voltage it should be about 14.5 volts after about 10 minutes with engine running.
The fridge would flatten the leisure battery, depending on the setting and type of fridge, after about 24 to 48_ hrs.
During a sunny day, Not on mains hookup, you should be seeing a voltage of anywhere between 13 to 14+ volts from the Solar Panel from your voltmeter and at night that voltage will drop to the 12.9 volts of just your Leisure Battery.
Does the Heater fire up? That runs off the Leisure Battery.
I think your reasoning is correct.
The Solar Panel should be connected to the Leisure Battery Direct. Hence the 12.9 volts via your Solar app.
The Leisure Battery should be connected to the Habitation area fuses located under the front passenger seat.
The Mains Charger is connected to the Habitation circuitry and hence Leisure Battery under the front passenger seat.
The engine alternator, via a special relay, is connected to the Habitation circuitry under the front passenger seat.
I think the Leisure battery has been disconnected from the Habitation circuits either due to all wires not being reconnected or a blown fuse between the leisure battery and the Habitation fuse box etc.
With the Leisure battery out of circuit the rear lights and 12v sockets would only work on Mains hookup via the charger or when the engine was running.