After trying a bunch of things, I finally managed to solve the issue.
I am not sure why it was like this, but while trying to reset my password from inside the web app, I got the error message "incorrect password" (even though it was the correct password, that both web app and desktop app recognized).
I then logged out of the web app, and did a password reset using this link:
https://account.battle.net/recovery/en/identify-account.html?requestType=PASSWORD_RESETAfter doing this, I could log in as normal into battlenet.
Doubting that anyone else reading this has this problem at this time, but if you do, just do the above!