If SSH works without a password prompt but rsync prompts for a password, the error could be due to how rsync is handling SSH.
Here is a quick fix for this issue: You should explicitly mention the path to your SSH private key using the -i option of the SSH command within rsync.
$RSYNC_PATH --no-compress --no-inc-recursive --delete --progress -aWe "ssh -p $RSYNC_PORT -i ~/.ssh/id_rsa" $RSYNC_USER@$RSYNC_SERVER:$RSYNC_DIR $BACKUP_DIR
If you have a different path for the SSH private key, replace "~/.ssh/id_rsa" with that path.