Based on what you can see in the Obama photo and your picture of the range, it looks to me, as if, he was standing in the little grassy spot, by the far right trap house, with the trap house at his back and is shooting, across the driveway, toward the T-shaped line, farther down the driveway.
In the Obama photo, you can see the driveway behind him and just the start of the curve of the driveway, in front of him. Which matches the picture of the range, in that spot.
He more than likely was dropped off at that point, took a couple of staged photos and left.
Could be, you might be on to something.
Im not sure at this point where he was standing, just know it wasn’t at any skeet shooting station.