Ok, maybe I oversold this a little bit. It's running smooth now, getting it to run smooth was not easy. I'm on Ubuntu. I spent a few days in a debug loop. Run steam from the terminal to get a log stream, keep an eye on CPU and GPU utilization and temperature, and futz around in the training range or vs AI bots (more "realistic" than training range). Identify which components of the system aren't performing up to spec. CPU running hot? GPU not being utilized? Steam emitting warning messages? If hardware all looks good, it's probably a software problem somewhere. Identify, then fix. Rinse and repeat until linux performance is in the same league as Windows performance.

Things I'd try:

1. Check in game graphics settings

2. Update graphics drivers to the recommended version (may be non-trivial, I had to update my kernel version)

3. Experiment with different proton versions, including proton GE

4. Experiment with different Direct X versions (in game option)

5. Make sure CPU cooler is running

6. Make sure GPU is being used

7. Use gamescope to configure a virtual monitor that exactly matches the capabilities of your physical monitor

I no longer support Blizz, so can't weigh in specifically, but: have you tried PrtotonGE? The are also the Proton forks, such as CachyOS's one that support wayland directly (which is in WINE, but not Proton yet) - might be xwayland relayed?

Also, try `LD_PRELOAD="" %command%` to disable steaminput, which can cause input stuttering after around 45min on some machines (such as mine).