I think we need to get to an understanding of what you have and have not done and what you are currently looking at because this is the timeline I see in your account notes:
Years ago… Attached physical Authenticator on account.
12/29 16:45 Detached physical Authenticator.
12/29 16:57 Attached the same physical Authenticator
12/29 17:59 Detached the physical Authenticator Authenticator - though self-help option
12/29 18:37 Attached Mobile Authenticator
12/29 23:59 Detached Mobile Authenticator - through self-help option
12/30 00:01 Attached new Mobile Authenticator
You said you have the Battle.net Mobile App, but have not yet logged into it because it is asking you for a security code, which you don’t have.
How did you set up the Authenticator on the account? Are you using the Authenticator App stand alone app? Or did you do it through the Battle.net App?
If you did it through the the Authenticator App, you should be able to open that and get the code when you are attempting to log into the Battle.net Mobile App.
If you did it through the Battle.net Mobile App the log in process opens up a browser on your phone that you log in through. At the bottom of the app, before you hit log in, there should be “Open Authenticator”.
You should be able to hit copy code there, and then swap back and log in. If you do this before the security code times out you should be able to just paste it in.