Here is my idea of a circuit to take 3 phase AC current directly from the
altarnator's stator winding, and use it through a transformer to boost the
voltage to the cooling fan motor.

This would add more power to the fan than a 12V system alone could provide.

The hard part would be finding an appropriately sized transformer core, and
determining how many turns of winding would work at the frequency range from
the alternator.