Why not make a game with 2 separated boards.
One player is white on one board and the other is white on the other board. With that no player will have first-move advantage.
Of course one rule to see who will win need to be decided. If the first one that mate the enemy (on any of the boards) win, being fast at mating would be more important than just mating, this is a not wanted thing.
Maybe do like that. If one guy win on both boards, he win the game. He both players win on one board, they play again.