The book discusses ways to fix the enamel on a button, but here is a way to polish up the metal without damaging the enamel itself.
Use a rotaty tool such as a Dremel or Flexshaft. These are probalby more espensive than this portable engraving pen.

This tool comes with lots of bits, mostly diamond, WHICH WILL damage enamel if used on it. The polishing tools I recommend are called 3M Radial Bristle Discs*. You do not need to buy a kit, as you only need certain colors. Mandrels to put these on will also be needed. The 3/4" size is good for your needs on buttons. You can use the yellow (80 grit), red (220 grit), blue (400 grit), and light green (1 micron). More than one head can be put on the madrel at one time, thus see what works best for you.

The blue and then the light green are perfect for gold and silver such as on Cloisonné wires. You can try these on other metals, but if you need something stronger than use the yellow and/or red. I usually use the yellow on copper or brass and then follow up with the red and then blue and then green. But note that the yellow may affect enamel (I cannot figure out when it does and when it does not, but usually it's fine - like it didn't affect my sample below); I'm not sure about the red as I just strated using it and do not have enough experience. It's possible that the yellow sometimes leaves a residue on the enamel that can be removed by an ultrasonic cleaner - I haven't tried this but will next time.
If you have quesitons you can contact me. Good luck.
Here is an example of what I did and why I thought to tell you all about this:
Original

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Polished

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* If you want to look to buy at RioGrande.com, you will have to start an account. As of today (11/29/2025) the price for a pack of 10 of each of the colors is $16.99 + shipping.
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