Here are a couple photos that might help nail down the issue. The bulbs and the connecting points are a non-factor, as the bulbs are removed and the LED wiring pushed past those points of contact, with exception of the reverse bulb. It wasn't a problem on the driver side at all.
In the photo below of the passenger (non-working) side, you'll see all the wiring is plugged directly into the housing coming from the driver's side.
In this photo below of the driver (correctly working) side, you'll see on the far left the Ground (three wires attached to the bolt) coming out of the housing. The passenger side does NOT have this. In fact, it seems like the Ground wire stops here and does not continue on to the passenger side.
I'm hoping these photos may help in relaying what the issue is. I may have not accurately explained before...or I'm not understanding the fine people that have chimed in. My apologies if I'm missing the explanation.
NicholasMav's post in LED upgrade tail light issue - '75 2002 was marked as the answer