I've been faceting since 1996 and have done a few repair jobs when
jewelers slipped when setting and when they then tried to fix it,
too. Although peridot can be difficult to polish, that is not the
problem here. The difficulty is that facets can only be polished well
on a flat lap when the facet's surface is absolutely flat to the lap.
Using a buff might take out a scratch. but it would simultaneously
destroy the flatness of the facet.
If the facet is small, as on your rose cut, you will also destroy
the sharp edges of the facet.
There are a few faceters who might undertake to polish a large table
while the stone is still set. If there are no prongs in the way, a
really good faceter could feel when the table was flat to the lap,
and use a pre-polish lap to remove the scratch and then, using the
same feel, could repolish on a diamond or oxide lap. If you're very
good at this, the table is repolished and the edges remain sharp or
even become sharper. However, facet meets can change if much material
has to be taken off. If the table is a rectangular step cut, this
isn't a problem, but on a round brilliant, the star facets are not
going to meet exactly if much material has to be taken off. So a
really good repair means recutting these, too, and that means
removing the stone from the mount and redopping it to get control of
where those facets are placed. Another problem is that the repolish
is likely to be better than the original polish, so for the very
discriminating you either have to polish at less than your best to
match what is there or repolish the whole crown.
Short answer for your rose cut peridot is, send it to one of the
faceters who specialize in this kind of thing and reconcile yourself
to taking it out and then resetting it. One of the best at these
repairs is Anthony Lloyd Rhees at
Tony can re-dop the stone and find the exact settings for each
facet using a mirror and a special grease. Each facet has to be found
separately so it is exacting work, but it is not all that expensive
for what it is. In some cases it might be easier to find a
replacement stone, but IDK about rose cuts in the size you might
Hope this info is helpful.