DIY polishing for polished cases

Hey OF, do you have any suggestions for a low-cost DIY method/material that I can use to polish off some tiny scratches on a polished side of the watch? It's not for anything that requires a professional service (a case of 'this can wait until I service this watch but just wanna get rid of this tiny scratch, if I can do so easily').

If you ever tried it, do you think 'Cape cod metal polishing' clothes do a good job?

Also as a feedback, I lately came across something called Bergeon Scratch Removal Pen, which does a good job on brushed surfaces but cannot be used on polished parts. In a way, I am looking for its equivalent for polished parts.

Thank you!

Thank you! Sorry for delayed details btw but here are some: the scratches I want to get rid off are on a thick and relatively straight SS lug.

Also see below for zoomed-in pictures (I think the one under light kind of exaggerates the issue but reflects all the scratches):

P.s. the lug edges of this one are relatively softened and thick, which is why I considered a DIY soft polishing in the first place. I don't think I'd even dare otherwise.

On this same topic, is it possible to polish a vintage KS without messing up the sharpness or form?

At some point after seeing so many badly polished ones, I started to think polishing is just better not done or rather impossible to do right on these watches.. but perhaps it's just idiocy/inexperience at play.. people ruining watches when they could just polish them right?

I did see two KS watches which had perfect edges and admitted to being polished.