I guess we can start at the battery itself and find the POSITIVE TERMINAL (+) and look for an IN-LINE ATC FUSE HOLDER.
It will look something like this...
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This ATC FUSE will be a LARGE SIZE rating fuse. It usually is right next to the battery but may be just under the frame or maybe inside a electrical junction box mounted to the frame. This FUSE is suppose to blow to protect your battery wiring in case of a DEAD SHORT on the main battery cable.
You want to try to find this first then we will locate the other TWO FUSES mounted on your 12VDC Power Distribution Panel where your converter/charger is located.
One way to test this is to measure your battery voltage without the shore power being connected. If your battery is fully charged then your VOLTMETER will read 12.6 or so DC VOLTAGE. When you connect the shore power cable this 12.6VDC should increase to 13.6VDC if it is connected all the way to the converter/charger chassis.
The other way to test it is to disconnect the positive terminal battery lead and see if you read the 13.6VDC when the converter/charger is running when shore power is connected. The down side of disconnecting the battery cables is you might spark it against something and that will blow fuses. Its best to measure thing all connected up.
Since you already know your ceiling lights don't come on without shore power connected this is telling you the battery 12.6 VDC is "NOT" making it back to the 12VDC Power Distribution Panel. You need to start tracing connection I guess to see where the 12.6VDC is not being read with your DC VOLTMETER. the first thing is should see is the IN-LINE FUSE we are looking for... If you find the ATC fuse holder shown above then you will need to pullout the ATC fuse and see if it is blown or not...
Roy Ken