First problem: Dim lights=low voltage on the 12volt side.
Did they do a deep draw down load test on the battery? Sounds like a bad cell. It should show about 13.6 volts under no-load conditions. 12.5 volts is low. That could be a surface charge that is killed as soon as you put a load on it. A bad (partially shorted) cell will make it impossible for the converter to charge the battery.
Second problem: Lights cutting out on shore power.
The thermal protection on the converter is kicking out. Was the battery connected when this happened? ((ON EDIT: I see that it was disconnected. That makes it tougher but it still sounds like thermal overload.)) The converter should be able to carry the entire 12volt load when on shore power, but if the above condition (bad cell) exists, the battery may be pulling so much current trying to charge that the converter can't keep up.
Electrical problems can drive you batty. Good luck.