In OR 2, indeed Maj Wheat's La Battalion did take a flag. But it does not say do to the fact the report is *Not found.
Col Radford captured 2 flags (69th New York), also said they captured Col Corcoran who was the commander of the 69th New York aka 2nd NY Militia. The 69th New York lost both of their flags as stated in OR 2 page 497.
"Colonel Radford, with six companies Virginia Cavalry, was also ordered by General Johnston to cross Bull Run and attack the enemy from the direction of Lewis' house. Conducted by one of my aides, Colonel Chisolm, by the Lewis Ford to the immediate vicinity of the suspension bridge, he charged a battery with great gallantry, took Colonel Corcoran, of the Sixty-ninth Regiment New York Volunteers, a prisoner, and captured the Federal colors of that regiment, as well as a number of the enemy."
Have a good morning,