The hint about the connector not fitting flush to the case was invaluable. Yep, using a lighted magnifier revealed “pocket fluff” packed against the back wall of the socket. I used a modellers knife to whittle a toothpick to a flat blade and carefully removed the packed lint with that. I was able to break up the lint with the toothpick enough to get it past the pins without damaging them.
I strongly warn against using any metal tool as any contact with the pins could result in a short circuit of the Li-ion battery, which would not be pretty!