I would put in 3 relays - coil powered off the TV's pigtail, and using the trailers battery to power the outgoing lights to the second trailer. put a fuse in each line (3) to the second plug. If anything goes wrong in the second trailer, it blows its fuse, and the remaining lights are unaffected as well as all the lights on the first trailer.
I check my lights at every hookup by turning on the headlights, and hitting the hazards. This runs all the lights on the trailer, and I can easily see if something is not working. I do that (walk around) at every stop where someone gets out. On my last truck I had a lighted 4 way plug, there were 3 small leds in the plug so you could see if the light problem was on the TV or trailer. I would use something similar of the back of your first trailer to simplify troubleshooting later on.
