I too have been having the same issue. The Battle.net application response when I press the Update button in the Battle.net app is “Please try again after logging on as an administrator.” This is for both retail and classic versions.
After some research and trouble shooting I found that the permissions on the /Applications/World\ of\ Warcraft/Data directory was set to 666, basically my admin account could Read and Write but not execute.
I tried launching the game manually but I received a modal window with an error of “Cannot start WoW. ClientHandler: initialization error: failed to create storage component.” I did a quick Google search and found this article https://us.battle.net/support/en/article/13934
Followed those recommendations to no avail.
I am able to manually launch the game now after opening up a terminal and running the following:
“sudo chmod -R 777 /Applications/World\ of\ Warcraft/Data”
Looks like a recent patch to the Battle.net application migrated some bad file permissions.
Mid 2017 iMac running Mac OS 11.6