I took more of a “big hammer” approach and just used epoxy to glue that ball joint permanently. I’m too impatient to fiddle with the shrink-tubing and I figured I wasn’t turning back at this point. I pulled the socket assembly apart, cleaned the ball joint and then covered it in epoxy. There was still a lot of play in the ball joint so I hung the whole assembly upside-down so it would come straight out of the housing.
Then I also added some epoxy into the joint from the top just to reinforce it:
I let it cure for 24 hours and the thing feels as solid as a rock!