It has been running flawlessly on Proton for years. Here's one way to do it without having to tinker with Wine guts or whatever.
1. Install Steam using your distro's package manager, e.g. sudo apt install steam
2. Download battle.net's setup.exe from their website
3. In steam, "Add non-steam game" and add the setup.exe
4. In steam, right click on the battle.net setup.exe and in properties, under Compatibility, check the box that says "Force the use of a specific Steam Play compatibility tool" and select Proton Experimental.
5. Run it, which will allow you to run the installer on the first run. On subsequent runs it will just launch the battle.net launcher, allowing you to run the game immediately.
And everything works!!! Of course, you can feel free to do it manually with Wine or using a different tool such as Lutris. To be honest, this works with almost any Windows game that I can think of.
