I'll add that in my attempts to get the rear right light to blink properly (with the engine running as that's only when it doesn't work, minus hazards), I've come to realize that it's probably a faulty light/wiring. The car throws an error code in the BCM whenever the light's blinker wires are attached that says -- Rear Turn Signal: "circuit short to ground" --. I tried googling that but I'm still not 100% sure if it's saying there's no ground detected (bad), or there's electricity coming through in the ground (also bad), and I've not been able to test it really as whenever this happens, the car cuts the power to the blinker right away (you can restart it by stopping the blinker and starting it again). I tried bridging the negative/ground of the light over to the other ground that's in that harness, but that didn't help.
Hopefully the seller will sort this out. I will say that trying to open the light is a no go, it must be glued down and then they put screws in to hold it together better. Can't argue the quality there...
Edit: In contact with seller now who wants me to try switching the lights around to see if it makes a difference (so it's not car-related), will try that and see. He also said to switch the resistors on them, which I'm not sure what he means as there's no detachable resistor as far as I can see. There's a black "bar" with two wires which could very well be the resistor, but it doesn't come off in any easy way without cutting the wires.
Edit 2: Looks like it's my car. I tested putting the rear right tail right on the rear left wires, and it works fine there. The seller has been really helpful to try to figure out the issue, so today I'll rip open the panel again and see if it's a harness-related issue and/or if I can find what's causing the short to ground.
Edit 3: Probably last edit. I gave up on making it work "factory style". I think the car might have had an issue on the right blinker for a long time, but for whatever reason, not everything triggers it. I ended up running a 12V cable to a relay that I stuck on the back of the headlight, using the flasher wire (purple) as trigger wire (all other wires should make sense). That "fixes" it. Not pretty, doesn't make me happy, and a relay is "clicky", but at this point I can't think what else to try except a potentially expensive dealer visit where they rip the car apart trying to figure it out why the car thinks there's a short. I hope no one else runs into this! If someone knows if there's a "silent" car relay, hit me up