Well, you have 2.5TB in /mnt, which is normally empty (it is only a place to mount external storage). Maybe you can move whatever is there to somewhere else, like an external storage? Also 3.7 TB in /var, that is probably ISO files and the like that can also be removed/moved.
Backing up to /mnt is a really bad idea unless you then move those files somewhere else. It would be better to mount some external storage there and back up to that.