If I was going to run just a single wire, I'd get a long drill bit (I bought a 3 foot long one from Home Depot), then take the wall plate off, put the bit at the bottom side of the box (outside of the box, not in it), angled as sharply as possible.
I'd drill down, hopefully within the wall and through the floor plate and into the crawl space. I'd wiggle it around in the hole while it is running, to make the hole a bit wider going through the floor plate. Then, I'd leave the drill bit in place, go into the crawl space, find the end of the bit and wrap a wire into the flutes of the drill bit for 5-6 inches. I'd be sure that the wire was small enough to completely fit in the flute. I'd tape the whole thing together with a single wrap of electrical tape for the length of the area where the wire is in the flute of the drill bit.
Then I'd come upstairs and carefully pull it back up, hopefully with the wire attached. If it was a smaller wire than needed for the ground, I'd attach the appropriate wire to the end of the pull wire and pull away. Then, pop out a knock-out from the box and thread the wire into that hole and screw it off.
If the wire doesn't come back with the drill bit, I'd sigh deeply, drink a beer, then put fish tapes through the bottom with a hook and from the top with a hook and try to catch the hooks and pull it through.
If that didn't work, I'd tear out the old box and then the fish tape from the bottom would be easy.