Yes I think he gets in right away. 500 goals and 1000 points would make it more obvious, but they will like that he won pretty much everything and for a hall of fame, Perry has been fairly prominent (famous) for a long time and memorable as one of the most prominent rat types ever. He's still very relevant now at 40 and whenever he is discussed for years now it's pretty glowing praise. His relatively fluky Hart helps a lot too. It's the hall of fame, not some legitimate ranking of the best X number of players ever.
I can't see any way he is left out, and given that he will likely be finishing his career (or close enough) playing relevant hockey on good teams, memories will be fresh for the committee and he likely gets in early on.