You definitely don't want to paint it, that will scratch off very quickly and look worse than it does now.
If it is important to you, meaning it really bothers you to have scratches on the top of the pup, then you have one of two choices: 1) sand it and polish it so you don't notice the scratches as much; 2) go back to the luthier who did the fret work and ask him to make it right (either put on a new top bobbin or replace the pickup).