Late reply but I only just found this post.
It's the framerate that's messing up your mouse, but using the in-game FPS limiter doesn't play nicely with Proton/Wine so here's my solution.
In the launch options for Black Ops add
DXVK_FRAME_RATE=125 %command%
You can replace 125 with whatever fps you you can consistently hit. In-game you have to turn off vsync and uncap your fps, but for some reason BO1 defaults your settings if you leave the fps uncapped, So instead use the console command com_maxfps 200 just make sure the number is higher than what you've set in the steam launch options.
Also if the game ever asks you to launch in "safe mode" click no or it will revert all your graphical settings.