Sounds like you want Looking Glass. It'll copy the frame buffer from one GPU to an application on the host, and deal with mouse input, keyboard input, and audio as well.
However, Windows virtual machines get detected by anticheat and you'll get kicked out of some multiplayer games. For some multiplayer games you still need to boot a full copy of Windows. If you set everything up right, that copy can be the copy you also boot in your VM setup, though, so you'd only need to boot into Windows for a few specific games.