Is there any way to allow a teacher to backup a course, but not allow them to download the resulting zip file? Maybe somehow prevent them from viewing the backupdata folder.
That's a weird request. Preventing teachers to download the contents of their own courses.
Well, that's just it. It isn't always their own course. Sometimes, they are hired on contract to teach an existing course.
Give them non-editing teacher role? I must admit that I would need to check if that restricts their file system access. The file area in Moodle isn't (by design) spectactularly secure once you have some sort of role in that course. So, if it *really* matters to you it might be better to restrict making backups altogether (to people you trust to delete them again)