257 chassis wiring

The pickup is secured with eyelets, one has the pickup wire soldered to the middle of it. Removing the pickup will break the fiber plate.  Since the wire is attached to the pickup, strip the insulation and slide heat shrink down the bare wire until it contacts the plate.  You can shrink it with a heat gun.

The wire from the pickup goes to one brush holder, the field wire goes to the other brush holder.  If it runs backwards, reverse the wires to the brush holders.

Isn't there a way to pop the fiber plate out w/o breaking it?

They are too brittle to remove.  The train tender sells just the fiber plate and I usually just transfer the pickup and name tag to the new fiber plate.  The new ones are flexible enough to bend when reinstalling.  I use non-marring pliers for the ends and trim the center one slightly and so it can be forced in.