I haven’t seen this episode of Seinfeld, but there is an opera called “Pagliacci” by Ruggiero Leoncavallo. It’s one of the roles Enrico Caruso is very famous for. It is pronounced more like “pahl-YA-chee”, but you’re very close. Unfortunately, I can’t tell you the name of the excerpt used in the episode.
BTW, I don’t watch television,and have never seen one episode of Sienfeld, therefore do not have clue #1 about this; but “Vesti la Giubba” is the aria from the opera in question.
In it the protagonist mournfully forces himself to don the costume of a clown, even though jealousy is poisoning his heart.