This works like that you descripbed.
A solution could be to move the player further then the other door when moving through the door. Like teleport him to in front of the other door. not inside it
Event - Player on collision with Door 1
Action - (move position-player moves to Door 2 + SOME_MORE)
Event - Player on collision with Door 2
Action - (move position-player moves to Door 1 + SOME_MORE)
Or set a flag 1 when the player has moved through the door and set it to 0 when its not overlapping door, and only let it move through when this flag is 0.
Event - Player on collision with Door 1
Event - AND MoveThrough = 0
Action - (move position-player moves to Door 2)
Action - Set MoveThrough = 1
Event - Player on collision with Door 2
Event - AND MoveThrough = 0
Action - (move position-player moves to Door 1)
Action - Set MoveThrough = 1
Event - NOT(Player overlapping with Door1)
Event - NOT(Player overlapping with Door2)
Action - Set MoveThrough = 0