OK, in boats I've known, and Journey On is no exception, the cabin lights are 12 Vdc and thus only operate off the DC electrical circuit. This is powered by the battery. When one is connected to shore power, and the (DC) battery charger is operating, it supplies power to the battery. The battery is still the source of DC power. Notice that when the charger is plugged in, the lights are brighter, because the charger has raised the battery voltage in order to charge it. It doesn't matter how the battery charger is connected to shore power, as long as it's output is wired to the battery.
Additionally, 12 V bulbs are designed to operate with ~12 V, AC or DC, it doesn't matter. 120 V bulbs are designed to operate with 120 V. Don't interchange them. If you put 120 V to a 12 V bulb the filament will blow, and if you do the opposite, you'll get no light.
If the lights don't get brighter, the charger isn't wired to the battery. I'm not sure what the main cabin light breakers are, but with or without the charger on, the cabin lights operate on the DC electrical circuit, connected to the battery. The switches don't change.
Note that when the engine is running, the alternator also charges the batteries. Also, it's normal to wire the battery charger directly into the AC electrical circuit, with it's own breaker. Safer, easier.
Sorry for the lecture.
Boris