The hyperfine interactions consists of an electric quadrupole interaction and magnetic dipole term. The quadrupole interaction stems from a nucleus deformation, which might not be easy to achieve with a bar magnet (if the carrousel is spinning at extremely high speeds, you could achieve an ‘averaged-out’ deformation as is the case for a real nucleus, but that poor child wouldn’t want to take a ride, I think).
A dipole interaction would be more realistic. For this, the charged ball would be placed at the edge of the carrousel, so that it ‘orbits’ around the center. The magnet could be placed at the center of the carrousel to be used as an analogue to the magnetic moment of the nucleus. However, the child would have to rotate the magnet opposite to the rotation of the carrousel, so that the relative rotation of the magnet and the ball is guaranteed.
The three contributions can now be reproduced.
For the orbital contribution:
The orbit of the ball around the magnet would cause this contribution, as already mentioned.
For the spin dipolar contribution:
This time, put the magnet on the outside and the ball on the inside of the carrousel. The magnetic field of the electron reaches the inside of the carrousel, which can be demonstrated with the magnetometer.
Finally, the Fermi contact contribution:
Bring both objects into the center of the carrousel. This changes the configurational energy substantially, but remember only a part of electrons actually are very close to the center.