Depending on how scratched it is, I might do nothing. SS is softer
than more of the stuff that goes into a sink. If the scratch pattern
is random, varied & wide spread it will hide future scratches.
My mom's house has the same SS sink for 40 years. It is somewhat
scartched but hardy more noticeable than when first new.
The appeal of SS;
minimal, it any, staining
more felxible, fewer china / crystal breaks
I assume you only want to polish the bottom of the sink.
If you must you, can probably use the method I used to save a SS pot.
I left it on the stove for 2 hours on high (by mistake) with two ham
I used liquid Comet (Softscrub would work).
I rigged up a polishing tool using a 4.5 or 6" sanding disc backer
(hard rubber with a threaded fittting in the back)
I put a threaded rod into the backer & into my electric drill. I faced
the disck with some drywall sanding screen (fine), a couple layers.
I squirted the Comet into the pot & went at it. Took about an hour but
those ham hocks were cremated onto the surface.
Are you sure this this necessary?
Try to avoid using a cleanser with chlorine beach, SS is sensitve to
chorline especially in a process like this. Do not leave the cleanser
on the sink when your not polishing, rinse well