Question: How did Hugo Strange find out that Bruce is really Batman?
Answer: Listen to the fourth Riddler interview tape. He did a psychological profile of Batman, then picked among candidates as to who could be Batman. He's also smart enough to realize only a rich man could afford all of Batman's gear and vehicles.
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.