OBR says when the ball is dead a runner may return to a base he left too soon if he has not touched the next base from the time the ball goes out of play. In your scenario, the runner crosses home plate before the ball enters out of play, therefore he technically has no “next base” since that refers to the position of the runner the time the ball went out of play (as stated above). In this situation the runner should be able to touch home, 3rd, 2nd, retouch 1st go back to 2nd then 3rd, and be legal and not out on appeal if the defense tries to do so.
One question that was left unanswered was where the runner was the moment the throw left the outfielder’s hand, if he is only awarded 3rd, we would be lead to believe he was between 1st and 2nd. It is a 2 base award from the time the throw left the fielder’s hand.