Yeah, I think so. As long as it's completely separate, then I guess you can run a new wire.
In fact, before you do that, just cut the brown wire loose from the harness side of the connector and see if everything starts working as expected. If so, then you can run your new wire. If not, then you know you got other problems.
The problems in America arenít from liberals or conservatives, democrats or republicans.
The problems are bigots, hate mongers, fear peddlers and opportunists. History has shown that those people are far more dangerous than a few guys who talk differently or wear funny hats.