First of all, have you actually observed this happening the way you describe?
If you examined what went on a bit more closely (slo mo camera) you would have seen the impact, in which the bat and the peg were in contact for a finite time. During the contact, there was some distortion of both, which temporarily stored some elastic potential energy. During contact, the batter could also have been pushing against the resistance of the peg. The flywheel would have gradually (over the contact time) accelerated and the bat would have slowed down. Once the contact ended there would be no further acceleration - if you assume the flywheel was rigid. If there were spokes involved the hub could accelerate after the contact time (catching up with the rim and the rim might have accelerated a bit as the peg straightened.
But, once contact had finished, there would be no more energy transferred to the wheel. There can have been no 'magic' store of energy; it must have all been in the strain of deformation of the bat and wheel.