Nice post - thanks for putting it here.
The 11th North Carolina Troops were originally the 1st NC Volunteers and fought at Bethel in 1861, their flag probably being the first CS battle flag to get a battle honor (for Bethel) added to the flag. It is a First National and was originally Co. E's flag.
The North Carolina Volunteers regiments were renumbered by adding a "1" in front of their original designations and they were changed from volunteers to "troops"; thus the 2nd NC Vols became the 12th North Carolina Troops.
There were also a few North Carolina State Troops regiments, also formed in 1861 like the volunteers, so this can get confusing.
I doubt they carried this flag at Gettysburg and probably used their ANV Third Bunting battle flag.