Whats the deal with black holes and "no information from inside the event horizon can leave"?
You are right in that, if you could send a message of any kind from inside a black hole then there is no magic required rebuild you from the information about you. If you could send even a single bit of information, then the black hole wouldn't be so black. You want to do this using the rotation of the black hole.

This doesn't work, for various reasons.


  Information has left the black hole; its mass, rotation, location, and a few other properties


The mass etc is a property of the black hole, not information travelling from inside the black hole. The reason is that you can't manipulate them in the way you describe, so they can't be used to send a message. 


  I then enter the black hole and construct another station


Here is your first problem. From the frame of reference outside the black hole, there is infinite time dilation at the event horizon. Hence you never quite reach the event horizon, nor enter the black hole. 

From the frame of reference of the infalling person, the event horizon is crossed. So let's somehow pretend that you are now inside the event horizon.


  Its purpose is to manipulate and sustain a small black hole "orbiting" the inside of the bigger black hole.


You can't orbit, you can only fall. The orbital speed is higher than the speed of light, so you will reach the singularity, as surely as you will reach next Tuesday. Your time has now been infinitely dilated, so events that take happen in finite time for you, take an infinite amount of time in the frame of the outside observer, so that makes sending a message using variation in rotation impossible.

Finally, from inside the black hole you can't change the total mass or rotation. The angular momentum of a closed system is conserved, so if you use your thrusters to change the direction of the small black hole you are carrying, the total angular momentum of the black hole remains constant. You can't change the total angular momentum with thrusters if you include both the angular momentum of the ship and the angular momentum of the gas ejected by the thrusters.

Similarly you can't change the total mass or location of the black hole from inside the black hole. And you can't do anything in a finite time from the perspective of an outside observer.

Thus no information can be encoded in the observed properties of the black hole.