So, in case anyone else has this part...there's no clip. Or not exactly. The thing sort of snaps together around the wires, then expands back to lock it in the hole. You could push and pull all day with it snapped, and they'd never budge. No feeding it through from either side of the panel or plate.
Not sure it's right, but what worked for me was squeezing the back with pliers, while using a screwdriver on the front to force it through before it could expand again. Then you can pop it open.
Engineer who designed that needs a good stomping. It also puts a hideous bend in your TM wires. Almost looks like you ran them through a Z bender. You wouldn't think that would be good for them, but it ran for 15 years, and still sort of does.
A look at it from the underside. Wires go in the notch, then make a crazy right angle bend to go through the panel.
That cannot be good for wire. Obviously that's where the cap doo dad fits.
Good clean fun right there.