Can we see some photos? What are the dots made from?
If the dots are soft - wood or plastic - you can drill them out. Use a
pin vise instead of a power drill. Mark the center of each dot with a sharp punch.
If they're hard - like real MOP or abalone - hit them with a sharp awl or punch and they'll crack, then you can pick them out with fine tweezers. If you mangle the holes, you can just drill out to a larger size and put bigger dots in.
Glue in the new dots slightly proud of the surface, then carefully file them flush with a very fine file. Hold the file parallel with the neck so it rides along the dots and use a very gentle touch, letting the weight of the file do the work. As you file, you'll feel a very sudden change in the "friction" of the file dragging along the dots - that's the point at which the teeth have just started to make contact with the neck finish, meaning the dots are flush, and you can stop. Then you can buff the finish. If it's not glossy, try a white scotchbrite to blend any file marks in to the existing sheen.
I like jnn pearl on ebay for dots. They ship from overseas so they take a while to arrive, but they have good quality and good prices. If you're just buying the one set, might not be worth it compared to a more expensive domestic supplier.