Rather than the Head Haunting passing a Treasure Hunter (by swapping positions, as stated in #3), the progress of the Head Haunting is blocked. This would allow the Hunters to strategically end their turn in front of the Head Haunting in anticipation of team mates defeating Ghosts or Hauntings. Thematically, this could be seen as delaying the Head Haunting by fighting it.
My thinking was that players would just place a player as a blocker for the entire game thus stopping the "clock".
What if you could block it to slow it down, but there was a penalty, like--if there is a player in front of the Head Haunting when it tries to move, it doesn't move, but instead frightens the player all the way back out of the house. And, if they had a jewel, they would drop it and it would go back into the closest room without a jewel (use lowest room letter to break ties).