First, a “flash shoe screw” is merely a piece of bent metal. I can’t imagine a cheaper part at low volumes. And even at high volumes the fabrication can be entirely automated. Assembly costs could be prohibitive.
But for an alternative, a twist-in tab would work with your parameters. Look at how toothbrush heads twist into their electric handles. Usually a quarter turn. Look at the really cheap brushes as they mold the mechanism into the plastic so it is “free”. If you need to conduct, you can insert mold metal bits.