What you said is theoratically possible, but how I see it, it has absolutely no strategical value. If the Bodyguard and Sane Doctor get to find each other in the game and agree to protect each other, the Mafia has freewheel to kill anybody else, not fearing to be stopped there.
So then you have this Doc-Guard couple that publicly makes out with each other, everybody knows about it, but soon there are only two of them left and they are outnumbered by the Mafia.
But like mentioned, the Doctor can be Sane, Insane, Naive, and there is no real certainty about it. This cain can be easily broken by the Insanity or Naivity of the Doc, or other counters, such as Mafia Assassin for instance.