Linguistic purists will argue that rats do not take an umlaut, so the word is properly Rattchen or Rattlein. In reply I direct your attention to other irregular declensions such as
I am firm, you are stubborn, he is a pigheaded so-and-so
or, as it were,
I am a purist, you are a spoilsport, he is a boring old poop.
The song they are singing is Ach Du Lieber Augustin. Do not ask me how I know this.