Error description: “Something went wrong with a file.”
Error code: BLZBNTAGT00000840

I think I have found a workaround: Set the Wine prefix’s Windows version to Windows 11.

Details:

• This can be done on the winecfg Applications tab, or by running winetricks win11. Note that it requires a fairly recent version of Wine (and winetricks if using that method) so it might require temporarily upgrading Wine/Proton in order to choose that option. Battle.net might lose your saved password if you do this, in which case you would have to log in again.
• Once the games are updated, returning to the previous Wine version should be safe, at least until a better solution is found.
• This seemed to be what solved it for me, but it’s possible that a side effect of my experimenting was the actual fix. I have reproduced this workaround since my original post, so I am now more confident that it works.
• Side note: I used GE-Proton9-22 in order to temporarily make the Windows 11 option available, despite the fact that Proton builds are not meant for use outside of a Steam-like environment. I’ll try a non-Proton build next time.

Alternative approach:

Mert Eren on the Codeweavers forum relayed another workaround. Summary: The problem apparently lies in the latest version of the Battle.net “Agent” program. If your installation is old enough to still have the previous Agent version installed, it is possible to remove the new one and block it from being reinstalled, relying instead on the old version to update your games and make them playable.

CodeWeavers are investigating for the sake of their customers, but have not yet announced a fix.

Ongoing discussions:

https://us.forums.blizzard.com/en/wow/t/couldnt-update-after-111-now-cant-install/2044960

https://old.reddit.com/r/linux_gaming/comments/1i1on1i/unable_to_update_world_of_warcraft_retail_error/