So, then there would be no running light wire running to the Mirror side LEDs. That would make them just Turn signal/Brake light. If you have a running light wire running to the Mirror side along with the wire from the rear brake light, then that is your problem. Just disconnect the running light wire and you should be ok.
Updated Diagram.
There is no mirror side running light, and no wire for it. Honestly - thank you for your help, but I think we got it early on. Just no time to prove it this eve. Truly - I appreciate it!