How should the disposal be wired without running neutral to ground?
2 circuits, 1 receptacle, 0 problems

The extra cable running to the problem receptacle is indeed your saving grace, as it provides an always-hot and neutral to go with the switched hot and neutral from the switched circuit.

So, to fix this, you need to:


Make sure both circuits feeding the receptacle-box (the disposal circuit and the dishwasher circuit) are off.
Take the two white wires in the switch box that are nutted to not-white-wires (the one going to ground and the other going to the bundle of hots), unnut them from where they were nutted, and nut them to each other instead.  This provides a neutral to the switched side of the receptacle.
Remove the nutted together bundle of white and bare wires in the receptacle box.
Pull enough cable from the cable lurking in the back of the box into the box and strip its jacket and wires back so you can use it.
Make sure the tabs on both sides of the receptacle device are broken off.
Nut the two bare wires in the receptacle box together and to a pigtail going to the receptacle's ground screw.
Wire the white wires to the silver screws on the receptacle and the corresponding black wires to the corresponding brass screws on the receptacle.
Button everything back up and turn both circuits back on.
Enjoy your new half-switched receptacle!