Shipon's is a good idea in light of fairness. Also, having 2 picks per round would reduce the number of rounds to 8.
So why don't we do 2 picks per round with shuffled places from the 2nd to the 8th round?
"And do not curse those who call on other than GOD, lest they blaspheme and curse GOD, out of ignorance. We have adorned the works of every group in their eyes. Ultimately, they return to their Lord, then He informs them of everything they had done." (Qur'an 6:108)