My best advice: keep your cool.
One of the reasons it is effective to copycat many times is because the player being copied becomes frustrated, and makes a dubious attack which cannot be copied. This attack is refuted, and the player is losing.
Just keep your cool, play well, play positionally, and at a certain point, he will lose the abililty to copy you because doing so will win you material.
I play 960 exclusively and I notice that there are A LOT of copycats there. What I mean is, at the start of the game and up to around mid-game, black will copy your same moves to the T. It really gets to me for some reason. What tips can you provide to stop that?