Since this is a new trailer I assume the LED lights are factory installed. Have you checked the battery voltage at the batteries first and then at the power panel with and without the pump running? Not sure how you are verifying that the batteries are charged and how much the voltage is dropping when the pump is running. You have a long trailer so the wiring could be having larger voltage drops. Is the pump on the same circuit (fuse) as the lights?
As others have commented,if the lights do not have drivers then small voltage variations can caused light intensity to change as the pump cycles. The water pump and furnace blower are the two highest current accessories unless you are running an inverter.
