I crunched the numbers on my method, and they give an average of 12.27, with the following distribution:
8 - 3.3%
9 - 6.7%
10 - 10%
11 - 13.3%
12 - 16.7%
13 - 20%
14 - 16.7%
15 - 13.3%
I don’t know if you ever came up with a satisfactory solution to this, but you could always roll 2d6+6, which has an average of 13, and simply re-roll all 16s, 17s and 18s as if they don’t exist. This would push down your average roll a bit too. It should cause your curve to lean a bit heavier...