You could shuck the drive and connect it directly or find a reliable USB enclosure.
Putting the disk in /etc/fstab by uuid should keep it mounted to the same directory at least, but I'll be surprised if the backup software properly handles the potential errors well. The following link has an example to remount the drive with udev when it goes away, and if the backup software isn't running as root you can mount with uid,gid set to the backup user (or chown an ext filesystem) but make the mountpoint directory 0700 owned by root to prevent the backup from writing to it while unmounted.
https://blog.backslasher.net/automatically-mounting-usb-driv...