ID a classical song from "An American Tail"?

If someone could help me out, I’ve got a classical (?) music question, regarding a song played in the background of the movie An American Tail.

The cleanest recording I could manage of it is here, though it’s a bit cluttered with some dialogue, a section of the US pledge of alliegence, and a few sound effects. :smack:

It’s not on the soundtrack, and a girl I knew once sang a bit of the same song, and mentioned the title and composer, but it’s been long enough that I can’t remember them.

Can anyone place it?

“Poor Wandering One” from Pirates of Penzance (Gilbert & Sullivan)

ETA: God I love that movie! The entire thing is burned into my memory. Your little snippet really takes me back…