The rules say that you pick up from where the game ended. This seems to mean that Dr. Lucky will start the game in the room with his killer.
If Lucky moves first [which seems to be possible, since the game picks up where the first game ends, so it would be Dr Lucky's turn to move], he immediately can attempt to kill his killer. Is this intended, or does the killer get an opportunity to move before Lucky gets his shot?
The rules state that 'The player who won the main game goes first.' and that Dr Lucky attacks when he 'enters an occupied room'. Technically at the start of this second phase, he doesn't enter the room. He is already there.
But even if you do decide that the player who won the main game is attacked immediately, he does have the opportunity do defend with the six cards drawn at the start of the Escape from Lucky Mansion phase.