I had a quite similar setup ~2 years ago. I purchased a basic version of Synergy [1], which lets you connect your keyboard and mouse to only one PC an still use it on both. There is such a low latency that I could play FPS on my Windows machine, while keyboard and mouse were attached to my Linux machine. The setup was quite bearable at that point, so I won't consider this a problem.

Base version costs around 30 bucks, I don't know if there is a free alternative though.

Disclaimer: I am in no way affillated with Symless or Synergy. I'm just a user.

[1] https://symless.com/synergy