I am working on what will hopefully be the definitive book on Georgia's Civil War flags for the last ten years or so.
Based on sources that I have, the flag was probably from Avery's 4th GA (later the 12th GA Cavalry). Those sources state that Clinch's 4th GA Cavalry sort of falls off the map after September, 1864.
The flag in question would have been one of the rectangular Southern Cross battle flags made by the Augusta Depot and issued to Wheeler's Corps in February-March, 1864. If the flag was marked with the units designation we cannot tell (none of those of this pattern that survive for cavalry untis are marked that I can recall).
The 10th Ohio Cavalry, which was part of Kilpatrick's Division, captured an unmarked flag of this pattern in Georgia (I forget the date of the capture) that rests today in a small county historical society in Pennsylvania. Perhaps this is the flag lost by Avery's troopers? More research is needed to tell.
I can state that I have nothing for definite flags of either 4th Georgia cavalry unit surviving today - and unless the one in PA pans out, that will remain so until something else comes to the fore.
Hope this helps.