Paul, I have a bit of an update. I just installed the latest version of BackupWP on a completely new installation, on a different server, running WP 4.0.1. The only things I have done to this site since installation are enable Multisite, install BackupWP, and activate it for only the main site.
This new install is giving me the same problem I was having on my other site (on WP 3.9.3), where the backup schedule page is only accessible through Network Dashboard > Settings, not through Site Dashboard > Tools, and clicking on the “Settings” link on the backup scheduler gives me the same “You do not have sufficient permission to access this page’ message that I have been getting on my other site. Seems like the plugin has some kind of compatibility issue with Multisite.
Edit: If I manually change the url that the “Settings” button sends me to from
https://siteurl/wp-admin/tools.php?page=backupwordpress&hmbkp_schedule_id=1416952795&action=hmbkp_edit_schedule&hmbkp_panel=hmbkp_edit_schedule_settings
to
https://siteurl/wp-admin/network/settings.php?page=backupwordpress&hmbkp_schedule_id=1416952795&action=hmbkp_edit_schedule&hmbkp_panel=hmbkp_edit_schedule_settings
(adding the network subfolder and changing tools.php to settings.php) I can access the settings screen.