Rather than poke things into the undamaged charge port of our battery, I built a quick-and-dirty mechanical duplicate:
The “center pin” is a snippet of what’s almost certainly 5/64 inch brass tube measuring Close Enough™ to 2.1 mm, with a few millimeters of 3/32 inch tube soldered on the end to simulate the nugget.
The aluminum rod has a 5.5 mm hole matching the coaxial jack’s diameter and depth, with a smaller through hole for the “pin” and a dab of Loctite bushing adhesive.
Then I turned the end of a 3/8 inch acetal rod down to a 5.5 mm bushing that completely fills the jack:
It has a 3 mm hole down the middle to aim homebrew shell drills directly at the pin, while preventing a short to the side contact.
The first test looked encouraging:
The nugget in the damaged jack is definitely larger than my soldered brass tube, but this was in the nature of exploratory tinkering while mulling the problem.