Hello Nathaniel
This may sound a bit radical, and I am surprising even myself here, but you might want to drag the stone behind your car. I have never done this before and it always sounds like a joke draging something behind a car. But what if you got yourself a good fresh wooden pallet as a skid and hooked it up to your car or truck on a nice quiet Sunday morning on the smoothest stretch of cement highway you could find. Get your friend to pull you behind and work the stone for a few minutes on the pavement as you go along. Maybe not hand held but with a jig of some sort. Anyhow I thought I would give you the honor of trying it out before me. Alx
P.S. don't forget the leather gloves, goggles and a crash helmet. alx

Before the advent of diamond plates a lot of sharpening stone lapping was done on cement blocks which are surprisingly have very flat surfaces. You have to have Popeye arms to do much though, but it will work.