I think there may be a typo in your message above. There are only 7 possible same sex opponents, and with 16 games to play, the best case scenario is that a player opposes 5 of them twice and 2 of them thrice. Do you mean keep the opposite sex opponents to 2 games?
I am thinking about this... It is not a problem to meet your "highest priority", it is balancing the opponents that is the challenge.