This member is "brainstorming" in another post regarding specific gravity. He is insistent if not self-indulging at the same time. I am not certain he will understand the magnetic force on a magnetic dipole depends not on magnetic induction, but the gradient of it, which is lacking in the center of the solenoid. Varying current is not going to vary this gradient so much. If you are close to the solenoid opening, then you do have this gradient. You can shoot something with a solenoid but the op wants some control.
Sorry to tell you; but you keep having it wrong...
The magnet is a real object with physical dimensions. As a cylinder it has height and volume. Its not a plane. That height makes it occupy a real space inside the solenoid and the top and bottom faces of it are not coincident with the solenoid center as will the ideal physical model you are describing. There is gradient from the top to the bottom face. There is also a force called gravity which will be pulling the magnet down and that will keep it off center if no energy is supplied to stabilize it there. On the other hand I never mentioned the center. That's you own creation; but that's Ok the magnet will stay at any position if enough energy is supplied.
Please stay proffessional, your insults will not make your opinions prevail.