When I mounted backup lights on my RV (and my boat trailer and utility trailer) I simply connected their positive wire of the LED back up lights to the 7 pin (center pin) of the 7 wire trailer connector.
On the RV I went to the junction box under the trailer, located the purple wire and spliced them together. Since the 7th pin was not being used and was cut short, I had to redo all the wiring to get a longer pigtail, no big deal.
As for the relay, why? Put the switch between the house battery and the positive wire to the lights, and don't use it while connected to the TV.
Or just use a jumper wire from the #7 pin (center) to the #4 pin and remove it when done.