You could indeed run Windows in a VM with PCI passthrough, and for a while long ago my desktop was Xen and I ran a Windows 7 domU which was attached to a second graphics card. Sharing a GPU at least used to be much harder; I think there's better options nowadays than before (paravirtualization-style GPU-command-level passthrough devices, and I assume some graphics hardware supports being split up for partial IOMMU passthrough in the way some high-end network cards do), but I don't know how they stack up for gaming performance.

However, the use case under discussion touches on things like handling kernel-level anti-cheat requirements, which is exactly the kind of place where I'd expect you to get in trouble trying to jigger around with virtual machines. Even before that point, I get the general feeling games and game platforms can get tetchy when you're not on Real Recognizable Hardware.