Lutris running proton-cachyos (native package)

How do you run this package on lutris or is that the GE-proton (latest) runner?

Use wine-cachyos in Lutris. There is a script to symlink proton-cachyos to Lutris runners directory however if you are insistent. See Add proton steam -> lutris symlink Option · Issue #78 · CachyOS/CachyOS-Welcome · GitHub.