Ugh! Blown facets are the worst! When you can't control them with your set-up, clone them out! If you need, with necklaces like these, take several shots with angles slightly different enough to get rid of the blown reflection to use as a source for cloning/compositing.
A tilt shift lens would be helpful for these images. You really want all of the necklace in focus, for a documentary shot like these. You could also focus stack. Helicon Focus is good software for that. Or mount it and shot it flat, parallel to the cameras sensor. If those were more of a creative, angled, detail shot, shallow DoF would be pleasing.
As far as more sparkle? Meh. Not needed. Nice clear focused detail is.
Ever notice stones always look more sparkly to your eye, than the camera?
I think it has to do with seeing them with binocular vision, and the camera is mono.
You could add sparkle flair in post, but, at what point is that cheesy?