That's because the copycat, in the first interaction, works exactly the same as the cooperator. And the outcome of that first interaction becomes more important

That's because the copycat, in the first interaction, works exactly the same as the cooperator. And the outcome of that first interaction becomes more important if there are less of them. The problem I see only pops up when there are more interactions - when the player knows the other will cheat, but still decides to cooperate.