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

A workaround that succeeded at least once for me: Use GE-Proton9-23, or kron4ek-wine-10.0-staging-tkg-amd64 from the Bottles project.

Details:

• GE-Proton can be either the one obtained from GloriousEggroll (which is made for use within Steam), or the one obtained from the Bottles project (for use without Steam). The Bottles kron4ek-wine-10.0-staging-tkg-amd64 download location is posted here. Bottles runners are described here.
• The next update after I switched to GE-Proton9-23 brought a new error: BLZBNTAGT00000846, which (according to Blizzard’s web page) means, “We tried to make a file writable but failed. Please log in as an administrator and try again.” I would usually say that doesn’t make much sense in Wine, but Wine 10.0 apparently includes some administrator access support, so perhaps that could help. I got around this by switching to wine-10.0-staging-tkg-amd64 from the Bottles project, and using Battle.net’s Scan and Repair option on the game. I’m not sure if the fix was the Scan and Repair, or the newer Wine build, or something in the Wine prefix getting reset when a new Wine version was detected, or some time passing between the error message and trying to launch it again (which could hint at a certificate timestamp problem), or some combination thereof. In any case, the next Blizzard update after that worked fine.
• I briefly thought that setting the Wine prefix’s Windows version to Windows 11 helped, but I have been unable to confirm it. For the record, 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.

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/blizzard/t/cant-updateinstall-due-to-blzbntagt00000840/53551

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/