Barnabas,
There is a traditional lapping technique, used by 'scientific instrument makers' and watchmakers, that might suit your need to finish the faceplate on the watchmaker's lathe.
Sorry, I can't remember the fancy name for it 
Basically, the workpiece [i.e. your new faceplate] is rotated by the lathe spindle, whilst a cup-like lap is rotated on an auxilliary spindle on a parallel, but offset, axis … the combined effect is to produce a near perfectly flat face, with swirl-lines [like a Catherine Wheel firework] just visible.
This is easily configured using a compound slide, plus vertical slide with a spindle; but I'm sure you could 'jury rig' something to use a Dremel tool, or similar. …. Most of the settings are non-critical: Only the relative alignment of the axes is important, and that can be tweaked 'til you see the right result.
Hopefully this rather laboured description will set you thinking, or perhaps someone will recognise the technique and offer a link to a video …
Meanwhile; you should find a lot of useful information in this book: **LINK**
http://www.opensourcemachinetools.org/archive-manuals/watch_makers_lathe.pdf
MichaelG.