Given that your group is concerned about injury with custom projectiles (and reasonably so), have you considered using craft foam for fins instead of cardboard?
Also apparently angling the fins a little bit creates spin, which is good for stability. Might be worth giving it a go.
Side note, have you seen MTB's video on making titan rockets? Might be a good source of inspiration, even if it doesn't fit the goal of interchangeable ammo.
They're all made completely from craft foam and pipe insulation, so the injury risk is pretty low. I don't think I ever considered cardboard; too thick, and it wouldn't make for as good of a seal on the barrel.
Some of the first ones have been made with tilted fins to induce a spin. But if KSP is anything to go by, excessive spinning hurts performance, so it's a little tricky making sure it doesn't go too far.
And yeah, I did stop by MTB's stuff on the way here. I considered hollowing out pool noodles with a Forstner bit on a drill press to get the inner diameter of the missiles to match the Titan's stock 40mm, but that seemed like an awful lot of work compared to wrapping foam.