You would change the Owner|UID to 570 for both. When you change the Owner of a folder it should change both of these at the same time.
Logically I would say no because your host server configuration was working for you before even though you are prompted for authentication whenever you want to install anything on this server. Typically you add any additional wp-config.php code like that when things are not working. Normally adding the define('FS_METHOD', 'direct');
code works on every/all servers and fixes any/all issues or problems, but for whatever reason it does not work on your particular host server configuration. You could check with your host support folks and ask them why the “direct” WordPress Filesystem API Method is not working on your website/server and ask them to change whatever needs to be changed on the server to fix that. Or simply just change the Owner|UID of the BPS Backup folders to 570.