As a faceter and cabber I would use Cerium Oxide in a water slurry.
I would mix both to the consistency of skim milk, keep the piece wet
with that mixture and as a carrier I would use anything from a piece
of leather to a tin lap. Of course you would have to adapt this
method to your Dremel, but it shouldn’t be too difficult.
Of course I would have gone through pre-polishing steps first. You
can’t go from “course” to polish without some effort in between. Many
faceters would go from shaping with something equal to 600 grit right
to the polish, that’s what I’d do too. I’m sure there are other ways
to skin this cat, this is just one.