Because he had no doubt that Tyrion was a Lannister...He did want that, but it's been so long since I've touched the books that I can't remember why he didn't go through with it.
It's a place with elemental zombies, trees and animals that act like wi-fi cameras, face snatchers and flaming arms... i for one would not be so sure death would end someone hatred.Tywin doesn't hate anyone now.