I'm pretty embarrassed by my own complacency, but I think I solved the issue.
The parking brake appeared to be jabbing at the cord whenever I pulled the brake release.
When this wasn't enough to disengage the plug entirely, it would weaken the connection, giving the appearance of an electrical short, via the loose plug.
All I have to do is re-route the cord and tape it nice and snug.
Well, that was a stupid mistake.