Revealing mistake: When Clayface disguised as Joker is holding Talia hostage in the movie theater, they are both standing on stage in front of the screen with a movie playing. However, they are not casting shadows on the screen. Also, no shadow appears on the screen when Clayface takes on his true form.
Batman: It was all a lie. There's nothing wrong with you.
Joker: Nice of you to say. But you of all people should know there's plenty wrong with me.
Trivia: Just before Protocol 10 starts, you end up in a giant fight with Joker and a good number of his goons. If you activate Detective Vision during this fight, you'll see that Joker doesn't have a skeleton. This is also true when he's holding Talia hostage in front of the move theater. This offers a hint to how the plot plays out.
Answer: Batman later in the game discovers that Hugo Strange has been supplying everybody with weapons, so he would have given Two-Face the vat of acid.