Williams flew for Marine squadron VMF-311. The neat bit is that another member of that squadron was John Glenn, and occasionally Williams flew as Glenn's wingman. Of course, Glenn was a lot less famous than his wingman back then.
I never did find the definitive answer I was looking for, but I did settle on a set of markings, although they were neither Williams's nor Glenn's. Neil Armstrong also flew the Panther, and I found a couple of good references for his plane, so I'm going to try to do that.