After much sleeping on it, I finally decided to make my own balcony door just like I made my own staircase. I used soft balsa wood, so that I can easily cut through it.
I used the bathroom door as a template for the this door, making it the same size. This was helpful because the bathroom door is the same width as the window opening on the tower wall. All I had to do was cut the bottom to make the window opening into a doorway.
I used the bathroom door trim to hinge this new door because it holds the hinges better than balsa wood trim. Since the bathroom door does not require hinging, I will use balsa trim for it instead.
The hinges hold well on the balsa wood, even though it's soft, but I didn't want to risk it on the trim part. The door itself is thick and has a better hold.
I then glued the hinged door to the interior of the tower wall opening. I am planning on putting a hinged screen door on the exterior.