I have been searching for the 44th USCT at Dalton, Georgia on October 13th, 1863. It appears that Three-fourth's of the regiment surrendered but so far nothing on if they lost their flag/flags. I would assume they did, since their Colonel was with them at Dalton Georgia.

Some books of interest,

"A Soldier's story: The Records of Hubbard Pryor, Forty-fourth USCT."

Trudeau, "Like men of war, black troops in the Civil War, 1861-1865."

I have searched the internet and google books and nothing mentioned on the flag captured.

I did come across the 8th USCT lost their "National Flag" at the battle of Olustee, Florida.
Source: Black soldiers in Blue: African American Troops in the Civil War Era. By John David Smith.

Have a good a good morning,

Shawn Prouty