No. Rotation battles don't work that way.
E.g., you have 3 Pokemon with Fake Out, Mienshao, Weavile & Hitmontop.
So Mienshao uses Fake Out, and of course Hitmontop & Weavile don't do anything. When Mienshao moves, that still counts as Weavile & Hitmontop's first move, as it's the first move of the whole battle, and therefore Fake Out won't work after one your Pokemon moves, regardless of which one.