If that the case I am not sure am buying the pick up of High freq Voltage, but seem the most logical. If you jumper across one of the glowing led does it go off and come back on with the jumper removed?
I like the battery in the radio but that does not explain the other 3 LED. If it is H.F. Voltage the LED will most likely not glow when you connect the mast as the light bulb (low resistance) will drain the voltage off.

Just reread your post. Attwood battery chargers, where did you tie the ground wires? If at the battery, never mind.

Should be able to pull the bilge pump wire off the battery to make sure it not the problem, I don't think it is.