Rear running lights go out when attaching a trailer without LED lights. Module?

Just flew to Ohio and Drove the vehicle back east, picking up a trailer in PA. That is where and when the problem was found. Ran home with flashers, which worked on the trailer and the truck (may be due to their intermittency). So, of course the place where I bought the truck "never" had this problem. Was told that the only trailer that they towed had LED lights on it. Vehicle only has 5500 miles on it. Began its life destined to be an ambulance....ended up with a custom aluminum flatbed. It was also mentioned to me that perhaps the LED lights on the bed may be improperly grounded....and are grounding through the filaments on the trailer lights or the test light when I applied it. Sounds plausible but hoping for something more definitive. Thank you for your help and interest.

Insufficient ground on vehicle at the tail lights....most likely! Do the front 'park' lamps go out as well?

No just all the lamps on the flatbed. Ground sounded the most plausible. Thank you, I will start there.

That's most likely your trouble! Run a wire from a known good ground to the bed and then to the lights...for testing. Bed is not grounded properly to vehicle chassis...i think...jumper cables are good for ground testing!