If so, how’d you do it? Or how would you recommend doing it?

Would an X-Acto knife work, prying off the felt-based bottom?

I have a very old one, and its magnet is extremely feeble. The penny shell is barely picked up, and insecurely falls off with the lightest of shakes or taps.

I’d like to replace it with a stronger magnet, if possible.

But how? Any insights appreciated.

...maybe try placing a much stronger magnet on the felt bottom of the square and leave it there for awhile? I don't recall much, but often this approach seemed to "recharge" weakened magnets.