Hi Zach!
Unfortunately, this does have a special feature: the pins on the “Pro Micro” side in the above come out and socket into springs hidden inside the black insulator (hence the “spring-loaded header with pins” from the title), and I need that specific feature.
This Japanese shop has them, but I don’t speak Japanese and I don’t know if there’s any way to get them to ship to the US: https://yushakobo.jp/shop/promicro-spring-pinheader/
Found some more info, and it turns out these don’t work the way I thought they did! Each “pin” is actually two legs spread out, which can be seen more clearly here: