I do not know what the flag looks like. Civil War units used a variety of flags. The 12th Bn. of Artillery was formed from the 1st Ga. Infantry which mustered out in 1862. The 1st Ga. had served in the Army of the northwest in the mountains of Western Virginia. The 12th Bn. served in the Army of Northern Virginia from 1864-1865. Therefore the surviving specimen could be one of any number style of flags.
Here is their assignment in the ANV
GEORGIA BRIGADE (CSA)
13th, 26th, 31st, 38th, 60th, & 61st Georgia Regiments
12th Georgia Light Artillery Battalion
Since units had close ties with their localities they sometimes had colors presented to them by their home community.
If the battalion had an Army of Northern virginia artillery flag it was probably the 3rd bunting issue.
As stated previously, it's quite hard to tell which flag survives. If you get to see this flag I would be interested in knowing what it looks like.
Perhaps Greg Biggs will see your post on here and comment.