Can I back up an entire NAS logical volume using Local Backup instead of backing up individual shares separately?

Yes, you can.

Create a share using a specified path (as opposed to the default path) pointed to /lv000x (the LV you wish to back up).
CONFIGURATION -> NAS Resources -> Create new share

Then simply back up this share.  As long as the share is the only data folder on the LV, you've essentially backed up the entire LV.

Note:  You'll need to create a new backup database or move it to a different LV.