The Puppet does not lie, so its abilities should work.
Abilities like the Knight ability still will not trigger for the Puppet, so they should probably be reworded to say something like "I can't die as long as I am Good."
My main problem is in the description for this game it specifically says "(but its other abilities like Knights immortality, will not trigger correctly)", so slayer working feels a bit weird. It would make more sense if it just said "I could not kill [number]".