Yes, you can use your flexshaft and polishing wheels for this, but it is even possible to do it with a couple of pieces of emery paper, maybe a 600 grit and a 1200 grit or 3000 grit or something close to these. A little water as lube on a piece of wet-or-dry paper. Set the paper on something soft, such as a mouse pad and move the set stone around until the scratches are gone. You can press as hard as you need to to make the emery paper conform to the stone’s curves. If the scratch is only on top, just press lightly to just work that area. For stones below Mohs 8 in hardness, oxide polishes such as cerium are more effective in polishing than diamond. Cerium is pretty cheap and you only need cab grade polish. You might even try going to a local optician to see if they would give you a little to use. You can try ZAM or rouge, but I’m not sure they would work. Linde A would work, if you had some. Diamond may leave you with a hazy polish, not sure as this isn’t recommended for feldspar, so I’ve never tried it. Just get a piece of felt or even pellon or cotton and apply the polish and a little water and rub briskly. A polish should come up quickly if the scratches are all sanded out first. If scratches remain, go back to the sanding steps. Lapidary polishing on cabs is not rocket science and should be pretty easy. It is trying to eliminate scratches on the tables of faceted stones with a flexshaft where jewelers get in trouble.
If you don’t want to fool with this, get in touch with the local gem and mineral club, as there are many lapidaries who could take care of this in 5 minutes and maybe they would like to meet a jeweler who could provide mounts for their stones. You might also find a source for stones for your work at a good price.