I'd probably try the sandwich approach as well. You could even do it on the mill to cut the center hole. Clamp the shim stock firmly between two solid and really, really flat plates (so the shim stock doesn't get pulled), and drill your hole. It might be better to drill a bit undersize then plunge a 3/16" 4-flute end mill to get the final i.d.
Remove, cut roughly to size with snips, then do the sandwich thing on the lathe, as you proposed. If you make the center bolt a close fit in the hole, you should get very good concentricity. I'd probably turn a stub arbor in place to be sure it ran true, thread the end #10-32 (#10 is only 0.003" larger than 3/16"), then assemble the sandwich on the arbor and crank down a nut.