On CachyOS:

I was not able to get the Lutris method to work. The installer kept complaining about a file system error and the Battle.net installer would freeze. I attempted this installation many times (~10) and eventually managed to install it without a file system error appearing, but even then, Battle.net would give either the "Battle.net Agent Went to Sleep" error or the "An error occurred while loading game information" error. I tried changing the Runner configuration to many other options than the default, but they all resulted in Battle.net freezing immediately after launch. I didn't try Jiloup's suggestion of using Proton Plus, though, so look at that if you insist on Lutris.

I was able to get the Steam method to work. Use Steam to run the Battle.net setup exe, and then re-target it to the launcher exe that is installed. However, the suggested Compability setting of Proton 9.0-4 still lead to the "Battle.net Agent Went to Sleep". Once I switched it to proton-cachyos-10.0-20251120, that error went away, Battle.net started normally, and I was able to install games. I then tried Proton 10.0-3, which also worked.

TL;DR: I'd recommend the Steam method, and Proton 10.0+