Regarding doing website backups. All backup plugins allow you to exclude folders/files from being backed up. There is no need to backup the /bps-backup folder (and a lot of other website folders and files as well) so you would just exclude the /bps-backup folder from being backed up.
Personally this is what we do with our website backups. We regularly backup the WordPress database since that is where all your content/work/Posts/Pages/Comments/Links/User Accounts are stored. We do not do any file backups since files can be easily replaced with new files, such as WordPress Core folders and files: wp-admin, wp-includes, wp-content. We do not store any database backups on the website. Database backups are immediately downloaded and deleted from the website.