OK...for one, you did not mention anything about moving your tank. That gets into a whole new process. However, as to adding live sand, wait until your tank is set back up again. Then add some to the existing bed.
Be very careful in filling the tank again after moving, so as not to churn the sand too much. How far are you moving it? Different room? Different house/apartment? If it is a small move, I would only remove enough water/rock to allow you to move it. Displace as little as possible.
I cannot understand why anyone would rinse their live sand. That will wash away everything beneficial in it. I would strongly recommend against that.
I think you are overcomplicating this. Leave the sand alone. Watch your feeding, make sure you have the right cleanup crew, and make sure you have enough current.