I do not think it is a wiring issue (but I could be wrong). I think the passenger light unit is defective.
I tested out the lights this morning with my other car, and this time the passenger unit running light did not work, but the stop and turn worked. Yesterday, on my other car (which is new and the first time I hooked it up to the trailer), the passenger unit running light worked, but the stop did not (I forgot to test the turn yesterday).
Anyway, I popped out both lights from their rubber grommets using a large flat head screwdriver. I switched the lights so the passenger light was now plugged into the driver side, and the driver light was plugged into the passenger side. The allegedly defective unit exhibited the same results on the driver side – the running light did not work, but the stop and turn did. The good unit now on the passenger side fully worked for run, stop and turn. Given that, I believe there is a short in the defective unit (I can see water inside the unit which is probably causing the short), so I am going to buy 2 new matching LED units (trailer is 10 years old) and see if that corrects the problem.
As an aside, it’s possible that my new car is not wired the same as my old car, and I may need to get that fixed.
Regarding the 4 pin connection mentioned above, this Brenderup has a 6 pin round, which I converted using an adaptor to a 7 pin flat. That conversion worked perfectly on my existing car. The new car came with a built-in 7 pin flat, so at least I have the same connection on both cars. I just might need to rearrange the wiring on the pins in the new car’s 7 pin flat.
For those looking to replace their Brenderup rear run, stop and turn light units, the cutout hole in the Brenderup is 4.5 inches, which is a common size for a trailer, and takes a “standard” rubber grommet into which a 4 inch round light unit fits. The units are sealed, and the bulbs cannot be replaced. The units have what appears to be a “standard” 3-prong connection, so the new lights can be installed by plugging the 3-prong connector into the new light unit, without splicing wires. The unit is popped out of the grommet with a flat head screwdriver, and is installed into the grommet by pushing the light into the grommet.