When attempting to use Git locally against Bitbucket, if an account is org managed, it's not possible to use your SSO password, as you get an error like the one below:

fatal: Authentication failed for 'https://<username>@bitbucket.org/<repoSlug>

In this case, you would have to use either an app password or an SSH key to proceed with the authentication attempt, as a workaround.

However, this feature request is to add the ability to use your SSO password for authentication locally.