Have you considered 12 gauge mops? I don't think you would need to use solvent. Just use as a "swiffer". I don't like breathing lead dust, so I would do it outside. Keeps it out of you house.
Yeah, I'm using a paper towel(so I don't ruin the mop head) wrapped around a mop and rodding them out now. Takes multiple passes to come out with clean paper. I was hoping there was an easier/more efficient method/tool.
I guess if you don't mind using solvents you can pick up a can of <45% VOC Brake Kleen and use the red straw in the nozzle. Either let it air dry or use some compressed air from a shop compressor to blow dry.