Ok, so, I may have found a solution to this. If I'm wrong, I'd like to pursue getting the part machined.
Turns out this actually is a standard part, it's just not that common. It's called an extended tip set screw.
https://www.mcmaster.com/#set-screws/=17td1yk
I've ordered some (box of 100 for 6 dollars, so let's hope I'm right) and at least know the threading and diameter will be correct.
Right now my only concern is that the tip on the broken part is a sort of black plastic, while the newer one appears to be all steel. Maybe that's just a matter of improved design, but I suppose I'll find out.
Anyway, thanks for the help. I'll update once the part comes in, and will potentially reach out for help should I need the replication. I still have the broken piece, so I'll be able to keep it as an example.