It's possible that Kidd's costume got modified by Connor. But as I said, his letter shows that he did not recognized them as Templars.
And if Kaniehtí:io was an Assassin, why didn't she recognized the threat right away? And how come the Assassin weren't the ones to take down Braddock? They were around way before Haytham's arrival, yet there was no mention of them doing anything to stop the Bulldog. Sure she had part in Braddock's death, but it was to protect her people, not for what the Assassin's wanted.