What happens when the Paranormal Investigator looks at a Doppelganger or Copy Cat who is now bad?
The Daybreak rules state the following for the P.I. (emphasis added):

If the P.I. sees the Doppelgänger that saw a Werewolf, he continues looking (he doesn't know what the Doppelgänger saw, and does not become what the Doppelgänger saw, even if the Doppelgänger saw a Werewolf or Tanner).
The Paranormal Investigator is on the village team, unless he finds a Werewolf, or if he finds a Tanner, in which case he becomes a Tanner and only wins if he dies.

So a P.I. who sees a Doppelgänger stays on the Village team, regardless of what happens to the Doppelgänger.