How to deal with a pop up drain install when nut doesn't sit flush with bottom of the sink?
The plumber's putty will not work, pull it all out of there please. The pop-up assy. should have included a fairly thick tapered rubber gasket to seal against the underside of the drain hole. This gasket has a flat back for the "crappy plastic" washers to rest against, those are actually friction rings designed to let the nut spin freely and press up against the rubber cone washer. I think you are missing the rubber washer.  

Plumber's putty should only be used to effect the seal between the upper drain trim ring and sink.