Hi Theodora,
I think I have the same problem with SourceTree. Yesterday it was no problem to push something, the app password is correct, but SourceTree does not use my-username@bitbucket.org for pushing how it looks like. Error message just shows use of bitbucket.org for the https path to my repository. Somehow I cannot see where I can tell SourceTree to use the path with the username, because the path in the push dialog window shows it with username.
In the meantime I saw the hints about SourceTree. I updated the version of SourceTree to the latest one. SourceTree stores old password under C:\Users\MyUserName\AppData\Local\Atlassian\SourceTree in passwd file. So, I renamed it and on push SourceTree ask for a new one. This time I use my app password. If I just refresh my login credentials under the options, it does not have any effect on my stored password for pushs before. The old and saved password in passwd file must be removed to make push run again.
BR Detlev