The script could be simplified and rewritten in the following way:
It makes sure all conditions are met before running rsync.
if [ "$(id -u)" != "0" ]; then
[ -d /media/Elrond ] || mkdir /media/Elrond
if ! mountpoint -q /media/Elrond; then
mount -t ntfs-3g /dev/sdb1 /media/Elrond
rsync -av --progress --delete --log-file=/home/trevor/backup_logs/$(date +%Y%m%d%H%M%S)_daily_rsync.log --exclude-from="/home/trevor/backup_logs/scripts/rsync-exclude" /home /media/Elrond/Backups/Thingol-Ubuntu/daily_home/
Schedule the script to run via cron with user trevor. When it runs it will re-execute itself as root. To be able to do that without having to enter a password you'll have to edit /etc/sudoers with visudo adding the following lines to allow user trevor to run the backup script as root.
That may then get around your encrypted homedir and rights problem.
Cmnd_Alias BACKUP = /path/to/dailybackup
trevor ALL = NOPASSWD: BACKUP
Edit: Just seen your post about using fstab to mount the partition - there's no harm in checking if it's mounted or not and mounting it if required.