Wow, Wandel, that is a tough task! I think it's good you paused the action and share the trouble.
I have sanded the white sides of two coffee house sets: The first one, because it was thickly lacquered, and while it did not look better afterwards, at least I had substituted the stickiness with a little hard wax in its place. Was it worth the time? Definitely not. The second one I really had to sand, because it had an extra coat of white paint. It was an incredible amount of work, like half an hour per pawn, an hour for each knight? Big pieces. Working carefully, one knight slipped from my hands and broke in two on the floor. Glue worked very well and I'm happy, but the knight with the broken jaw still looks mad at me with its scar. In the end, it was worth it, because I have the natural wood back and waxed or oiled it, a joy for me to play.
I am biased about your set- which you have presented here before- I like it as it is, colour and form and I would just stop and try to recover the bishop to its original shape and colour as good as possible. If you continue, to get into the crevices, I just folded the sanding paper. For restoration, I can only show you my bakelite Varna knight as an example: Clay was too wet, wood wax was too hard, so I took plasticine rubber for children, mixed the colour and restored the missing jaw. You can try this and then may have to apply a coat of lacquer to maybe achieve an intact 'crust'. My Varna knight I store with all the other pieces but in a tight jar, because it would lose form on impact.. 
Of course there should be professional ways to restore wooden shapes with some sort of wood paste.
For the finial: no metal, I think glue will do.
Okay, I’m pretty horrified with what I’ve done here. I acquired a ’50s–’60s Soviet chess set recently which I’ve dubbed a ‘Droplet set’ because the bishops have these finials that look like droplet impacts. I found 38 sets like this on Etsy, some of which feature kings with droplet finials as well.
Anyway, as the design didn’t seem to be super rare and I got it for a good price, I felt fine altering the pieces. I wasn’t a fan of the mocha color of the white pieces so I decided to refinish them, and I figured I might as well alter the shape of the bishops a bit so they look a bit more elegant. I was going to alter the queens as well, as their finials are comparatively too big. But that’s another matter.
So I got started on a bishop, and while trying to get a more gradual shape to the droplet, somehow the thin part suddenly became really thin even though I didn’t actively sand that part. And yeah, the finial just snapped! It’s a jagged break so I was able to put the droplet back on top without it falling off, but I’m obviously going to have to glue it down. Below is an intact bishop on the left, and the awfulness I created on the right.
So now I have two questions:
Below you can see the shape I was trying to go for. I honestly have no idea how I managed to sand so much off the wrong parts. I tried to sand just the bottom of the sphere to get a more gradual curve, and the sphere ended looking blocky on top so I tried to sand that back into a sphere. A marginally smaller sphere than before, yes, but what I ended up with is nothing like a sphere.
I thought, how difficult can it be? It takes a long time to sand material off, so I will just gradually alter the shape a little bit. An hour later, I’ve ruined a 60–70-year-old chess piece.
Oh and one more question. How do I sand the grooves in the base to remove the color there to make sure I retain the same amount of detail once I apply a finish to the piece?