But the joke here is clearly about how awful summer stock theatre can be, with Cathy specifically citing here the odd casting you sometimes get when theatres have to cast a core company for an entire season. Later in the song, she also mentions that she "plays Anita at the matinee." With the way that the role of Cathy is often cast, with white actresses, Cathy is just as inappropriate for the role of Anita as Carl possibly is for Tevye and/or Porgy. She also sings about being roommates with "a former stripper and her snake, Wayne." Now, Cathy doesn't mention what roles the former stripper, with or without her snake, is playing, but the image that conjures up is a hilarious disconnect between the performer and the types of productions one might normally find in this kind of summer stock.
The joke you reference isn't about Carl. It's about summer stock theatre. And, frankly, I always find it really hilarious.