I wouldn't leave soap on the chain without a rinse. If it gets wet you'll start washing off your lube. I also believe Dawn would work better on liquid oils rather than waxes especially with just a wipe. The main effect it would have on a chain lubed with White Lightning would seem to be where the friction would actively mix it with the wax, right where you need it the most.

Solvent might be safer than soap for the environment because you don't have a lot of dirty water to throw away. You can reuse your used solvent by letting the dirt settle out and when it's too dirty you can take it to a recycling center or burn it.