I’ve probed the wiring extensively to ensure there are no shorts
This thing's an enigma.
That "purple wire" is not what it seems. Something is not right with that (and other, evidently) connections.
You need to dig deeper.
Visually inspect the relevant wire(s) for ANY signs of being changed since leaving the factory, along their entire length.
ANY splices, aftermarket connectors, electrical tape, etc. are an indication that someone's ****ed with the electrical system previously.
You
may have two wires connected together somehow, e.g.,
- previous owner's sloppy splicing
- worn insulation allowing two wires to come in contact
Or, you may be the victim of some other, prior changes / "fixes" to the vehicle wiring, e.g., the installation of a car alarm or remote starter.
I would begin by confirming the continuity of the wire(s) (purple, etc.). This can be done any number of ways.
In the past I've used a small, 9V battery to test individual wires. I'll connect the 9V battery "-" to vehicle ground and "+" to the wire under test (e.g., your "purple" wire you've mentioned) and then go to the OTHER end of the wire and verify that IT measures +9V, and NO OTHER wires measure voltage.