Thanks for the help, everyone! I looked up how to remove the top cover properly and found there was a little lever sliding over at frame 2/3 which was preventing advancing the lever. I just taped it down with some gaffer's tape and problem solved -- advances through the full S-36 range now! I suppose it's not fully "fixing" what's wrong with the camera, but it does solve the problem and allow me to load it up with film and start shooting. I even cleaned up the viewfinder and rangefinder windows while I had the top cover off and everything is nice and clear.
John, thanks for the guidance to show me where to start looking! This isn't my first GIII or Canonet, but I was stumped as to why it just wouldn't continue advancing since it wasn't frozen solid.
Time to load some film and see how it does!