The lights, and the fridge, are controlled by 12 volts. Your A/C and perhaps your TV as well may be on 120 volts.
Your water pump is also 12 volts. It is also a heavy load. It sounds to me like you have a loose connection, perhaps either the positive or negative connection to your 12 volt panel box.
You could try a little experiment, shut off the water pump and the fridge, then go through the unit and turn on every light. If nothing happens, let the water out of tap so there is no pressure in the water line, shut off the tap, and then turn on the water pump and let it build up pressure. Again, if nothing happens, shut off the water pump, drain off the pressure, and start the water pump and the fridge at the same time.
And do this without being connected to shore power first, then try the same while connected to shore power and see if there is any difference.
That might give you clues perhaps. It will most likely be one of those heavy duty 12 volt connections is loose. Good Luck.