The following is from The Atlanta Constitution, 20 January 1890, p 5, "Paul Jones Semmes"
"Here in an old scrap-book is a yellow and worn piece of paper - the copy of a letter from Major S. P. Hamilton, of Cabell's division of artillery, Longstreet's corps...It is dated...July 26, 1863...addressed to P. W. Alexander, and forwarded with his correspondance to the Savannah Republican...Major Hamilton says: "I was standing at the gun...when I observed a wounded man being borne from the field in a blanket. By the number of attendants I soom perceived that it was an officer of rank, and in a moment afterwards recognized that officer as General Semmes..."
The article clearly indicates that Hamilton spoke with Semmes for a few moments - long enough he would have seen a flag, if that was part of the General's litter - before Semmes was removed further to the rear, where his brigade surgeon got to work. Based on this article, I think the flag as litter is post war embellishment.
The surgeon is identified in the article as Dr Todd, supposedly a brother in law of President Lincoln. I can't vouch for the Todd connection, but George Rogers Clark Todd, b. July 1825, was one of several of Mary Todd Lincoln's immediate family members serving in the Confederate army, and he served as a surgeon. [Todd info: reference "Donna McCreary as Mary Lincoln/Todd Family Genealogy Information", http://members.aol.com/beaufait/biography/geneology.htm]