A great suggestion.
Here in the US, there is a variety of very fine "open coat" sandpaper, which is light in color, and which does not shed particles. It's the kind of thing used for sanding automotive finishes and the like without clogging up.
3M makes one variety in sticky rolls, which may be too long to be practical unless you happen to have some around anyway (it's pretty expensive compared to regular sand paper), but it or something similar should also be available in small packs of sanding disks. I have a roll of 3M "Tri-m-ite" in 220 grit that works very well, though its stickum is minimal and you have to be pretty careful not to handle the sticky side before putting it on. I just put some on a few caps, and it looks as if it should work just fine.
It may help to degrease the cap a little first to get rid of existing finger grease before sticking.
If you can't find what you need at a hardware store, try an automotive supply store that sells paint and finish products.