Anyone have any advice on how to restore stainless exterior trim from fairly light scratching and scrapes?

I have been working in my Gilbarco 996 exterior trim lately but have been unable to get the polished, fresh chrome shine. I could easily buy aftermarket replacements, but want to reuse these as they are the original ones and want the satisfaction of the sweat equity.

My original understanding was to go over it with a nice wet sand between various grits to get the scratches out before giving it a good polish. I was using some of my automotive chrome shine with both a hand polish and later with my dremel. Unfortunately my test piece has not responded well to this. While the scratches have somewhat come out, little has changed to the finish beyond clouding the metal a bit. One trim piece had some flaking top layer, so I am hesitant to go at it again with another go of sandpaper.

Has anyone else done something that worked for these?