Justin If I remember correctly it was a problem due to Battle.net.exe and/or Agent.exe. There was kinda the same problem on Windows tho. Where Battle.net Launcher fucked with performance in games.
Solution back then on Linux was killall Battle.net.exe Agent.exe
Or just use the setting in Battle.net Launcher to close itself after launching a game, that kills Battle.net.exe but not Agent.exe IIRC.
And about OW in general. For me, it runs perfectly fine on Linux. And on Solus it feels even more smooth. My mouse feels better than on Windows to be honest. I do have 1x1 pixel movement on Windows, no skipping etc. no accel. ofc. But still, it feels somehow better on Linux/Solus for me. Only thing is, when starting the game first, it drops on frames for couple of seconds, because if shader caching or whatever. But thats about it.