Another morning of finger-plane work took the edge of the back to within a half-millimetre of the desired thickness. Other than the perimeters, both plates still have far too much material. However, once the perimeter groove is cut, I'll have a much better thickness guide.
My notebook says to route the perimeter "using a guide". Of course a thorough search of the shop did not turn up an appropriate guide, nor does my memory tell me what it should look like. So I made a new one from a dowel clamp. Since one only gets one chance at this, setup is critical. Using analog calipers and thickness gauges, I got pretty close to the proper offset and depth. Unfortunately the throat dimension of the drill press is a bit less than required to fully guide the workpiece around the guide pin, especially at the "C" bouts.
Since it was getting close to lunch time, I put this challenge aside until next week. I'll bring my digital calipers and have another look at the clearance problem. Maybe with another week's thought the solution will be obvious!