You could carefully cover the white soles with painter's masking tape (also called FROGTAPE™) and remove the tape after you finish.
The masking tape will prevent the polish from penetrating through. It can be removed without any chemicals as it holds by a liquid-tight adhesive bond.
The easiest way to disable the LED lights in children's (or adult's) shoes is to not plug the charger into the battery and let the battery go dead.
The battery can only hold a couple of hours of charge for lighting up your way.