So basically if the people that wrote the trojan made it to hack a wow account (not sure if that is what it was designed for) then they also would have a script which would automatically log onto you battlenet account and disable the authenticator all in a few seconds of getting the code.
As for trying to log in and getting an error and immediately on the second try calling blizzard or believing your account has been hacked, personally I have entered wrong passwords more than once on my WoW accounts and so getting it wrong twice would not be a red flag.
I do know that I just added an authenticator just last week due to my virus scanner finding a trojan on my machine which I instantly deleted and then waited until the authenticator was in place before logging onto WoW (I had the autheticator from Christmas but had not used it) .
This was the first time in many years that I have gotten infected, I never open attachments etc and not 100% sure where it came from, only possiblity is that one of the other family members used my machine (they are banned from it but for some reason my maching does not always prompt for it's password when it comes out of screen saver mode).
Overall, security is always only one step away from the hackers, sometimes the hacker is ahead and sometimes the security, just a game of cat and mouse.