I agree with all the above, especially testing with an actual volt meter at the battery. But you don't really need to disconnect any battery wires.
Set your meter on DC volts if it's not an auto ranging meter then set it to 20 V DC.
Put the red test probe on the battery positive (+ or POS) terminal.
Put the black test probe on the negative (- or NEG) terminal.
Read the voltage. This reading read be in the 12.xx volt range.
Now plug the trailer in and take a second reading. This reading should be in the 13 - 14.xx range if the battery is charging.
If the battery voltage is the same whether or not it's plugged in, then it's not charging.