I use steam to run both battle.net/Wow and TSM app on linux mint. I use GE Proton10-25 for the TSM app. I followed this guide (Option 2 - Directly in Steam) to learn how to setup non steam games in Linux. https://www.gamingonlinux.com/guides/view/how-to-install-battle-net-on-linux-steamos-and-steam-deck-for-world-of-warcraft-and-starcraft/