No. 6 at Country Club of Charleston is not a Biarritz hole and never was. It is a unique Seth Raynor design, a one level green at the end of a long par-3. The plans for the course and a drawing that was made by a club member and a appeared in a local paper around the time the course opened show a green much the way it is today.
Mountain Lake and Dedham's Biarritz were originally one level greens. In the case of Mountain Lake, a severe ran up to the green but there was not swale. I'm not sure if those should be considered a Biarritz or not. Brian Silva converted Dedham and Mountain Lake to the typical Biarritz design.
At the Apple Nine, part of the Lyman Orchards Complex in Middlefield, Connecticut, there is a Biarritz green on the third hole, a par-3, designed by Mark Mungeam. The course has 7 par-3s and 2 par-4s, including a Short, Punchbowl, and Redan. A ton of fun to play, lots of kids tee it up there. Greens run at about 8 feet.