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?
"O people, we created you from the same male and female, and rendered you distinct peoples and tribes, that you may recognize one another. The best among you in the sight of GOD is the most righteous. GOD is Omniscient, Cognizant." (49:13)