Per the instructions for BLZBNTBNA00000006 (BLZBNTBNA00000006 - Blizzard Support) I removed the battle.net tools folder. This only made the battle.net app not load properly and send me back to article 18771. Then I tried to uninstall, but Windows uninstaller couldn’t find it because it was looking in C:\ and it is installed on D:. I tried to just re-install in hopes it would overwrite the app currently installed, but the installer only gave me an error that “Oooops, something went wrong”.
Update:
I was able to get the battle.net app to update by going to my GeForce Experience app and launching Overwatch with that. When I tried to open Overwatch that way, I was automatically prompted to install the battle.net app. After the reinstall, everything appears to be working.