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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...

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