I finally got the sockets opened and so I made and inserted extra contacts for the heater pins out of copper. Being that I didn’t have any copper sheet on hand, I cut up a piece of copper pipe, flattened it out, and then cut the rough shape of the contact. Afterward I bent it into the correct shape and drilled holes for the connection screws. They’re a little rough, but they’ll work.
A word of warning: This takes way too much effort. More than half the contacts I made, probably closer to 2/3, were unusable and were scraped. As rare as these are, I probably would have been better off trying to find ones that actually had all four contacts.