Thanks for taking the time to do such a long reply.  So many things to pick up on........ But first. Are you making the outer cover for your box?  Looking at yours prompted me to go to the Mets website and look at it again.  I had forgotten that it has an outer cover that's a backgammon board. I actually thought the backgammon outer was a totally separate box way back when I added pictures to my Pinterest board !!!!!

Understand your comments about multiple layers, and yes I've glued my 'finished' registers in after - leaving the inside edges unfinished for the glue. Though as I'm not adding thicker pigmented 'lacquer', I can usually sand or plane the registers to fit and just put a bit more poly on or even wax polish. I tend to glue my mitered registers in and then put the lid on the little box while it dries which usually results in a good push fit.

Interesting you mention luthier timbers - I've used them for sound boards in music boxes. A friend who makes amazing ukuleles was showing me some Aussie desert timbers that a guy down here impregnates with epoxy in a vacuum chamber to create a hard very stable wood. He sells them world wide to luthiers. Oh and the friend also has a small humidity/temperature controlled room where he keeps his timber and work in progress!!!!

Yes basswood/lime is good, but can tearout.  I often use it for light coloured drawers. One of my favourite tools for lips and rebates for stringing corners, is the tiny Veritas shoulder plane - it's magic!!.  I also have their tiny plough plane which is another joy to use.  

An off-shoot of my veneering is that I often make my own plywood - e.g. three ply for tiny box bottoms. An idea I got from Andrew Crawford. Though I am starting to buy and use laser ply which you can get with a figured outer veneer. You can get it in packs 600x300mm which is useful.  
Martin