Non-stock wiring is always a gamble to diagnose remotely, too many possibilities, but here goes anyway.
If one were to ASSume that the six way conversion was/is done correctly and still functioning properly, then you've just described a broken ground circuit. If the plug seems to be the issue, that particular pin (and the wires attached to it) should warrant particular attention. If it's standardized, that pin should be at the ten o'clock position when viewing the truck plug, and the two o'clock position when looking at the trailer plug. ASSuming that is, that the "key" is at the top, and the trap door hinge is at the top. Of course, don't ASSume anything, but it sounds as if you've already verified the truck plug.
Side note- Back up lights on a six way concerns me a touch... I've only seen that on home spun wiring. Normally you would have
Left turn
Right turn
Tail
Ground
Electric brake
Constant power.
ASSuming that the wires are in a standard pattern, the back up lights would have to be in the center (replacing the constant power wire) or in the two o'clock position on the truck plug (replacing the electric brake) Watch out for that wire if you tow anything with brakeaway brakes or other onboard batteries that need charging (Popup campers come to mind).
Absolutely nothing wrong with using that as a back up light on the trailer, ASSuming that you aren't obligated to having a brakeaway device or electric brakes. No fun at all backing over stuff if you can't even see the carnage IMHO. Just be weary of hitching up to anything with electric brakes or a battery charging provision. Obviously the safety issue of not having brakes that you prolly should have, but also electrical issues from popped fuses right up to the possibility (especially with a weak plug) of having lots of brakes when you're trying to back up.
