MEMPHIS DAILY APPEAL [MEMPHIS, TN], December 3, 1861, p. 3, c. 1
Black Flag.—The Concordia cavalry, who carry a black flag, were in town yesterday.
MEMPHIS DAILY APPEAL [MEMPHIS, TN], December 5, 1861, p. 2, c. 6
The Concordia Cavalry, Capt. Benjamin, left their encampment at this point yesterday on the Magenta, for Bowling Green, Ky. They bear in their midst a large sized black flag on which appear, in bold relief, death's head and bare bones. These Concordians go to expel, not capture, vandal invaders of their homes and firesides, and they will make their mark.—Concordia Intelligencer, 29th.
Dr. Art Bergeron's "Guide to Louisiana Confederate Military Units 1861-1865" (LSU Press, 1989) identifies this unit as Company F, 1st Louisiana Cavalry Regiment (Colonel John Scott's regiment) commanded by Captain Joseph Benjamin. Bergeron's "Guide" notes the company to have been recruited from Concordia Parish, Louisiana. The regiment was organized at Baton Rouge on September 11, 1861 and sent to Nashville. They traveled by steamer to Memphis, took the train to Nashville, and then rode to Bowling Green.
I know this doesn't answer your question about the 1st La. Cavalry's regimental flag but thought you might enjoy seeing this, if you haven't seen it earlier.