Have a look at bb-includes/functions.bb-template.php you’ll see the functions function get_post_author( $post_id = 0 ) and function post_author_link( $post_id = 0 ).
You’ll need to recreate those in a plugin or in your functions file to use $user->user_login instead of $user->display_name.
Have a look at profile.php in your theme to see how it is generated there. But remember that on post.php you’ll have to pass in the ID of the post to get the username otherwise you’ll be fetching details on the logged in user rather post author.
I hope that helps, good luck.
An easier way would be to take away the ability to allow change of display name. Then it will default to username anyway. Have a look at topic/help-with-basic-command-to-add-new-profile-fields if that’s the avenue you want to take. If you do, don’t forget to set everyones ‘nicename’ back to what their username is.