re: Small box

Never had to deal with the ends fitting nicely 🤠

I can see why the string/berry technique is common, just avoid the tapered fitting by adding a joint element that hides the detailed misery!

Are you hammering flat the wire, or just laying it in a groove and then sanding the surface flat? 
Either way, you can also get decent tapers from simple diagonal wire cutters which "pinch" the wire apart leaving a tapered point versus shears (flat point).

Copper is also very malleable so if the wire comes up a tad short, it is easy to stretch (unlike wood 😩)

For shaping the tip on a sanding block, the only thing I can think of would be to use a scrap of wood with a groove cut for the wire to hold it securely. Press in the wire so the end  is even with the wood scrap, then drag the wood over the sanding block to taper. If you know the required angles, you could cut those angles into the wood as a guide.