"Puss" and "Pussy" in this song refers to the hare; the term was commonly used by hare-hunters, so there is no mis-hearing involved.  I've always rather assumed that Monigan was somebody Terry Woods knew, and that he was having a wee joke at his expense involving another common meaning of the word "pussy".  Conjecture only, of course; a deliberate double-mondegreen is an interesting thought.