First you need to determine what the sink is actually made out of.

This is an after-picture after using 800 grit sandpaper.

If nothing else works, maybe make the best out of it, grab a lighter (or soldering iron) and just burn a cool pattern on it?

Try rubbing with mild abrasive such as toothpaste - it'll take a while if it's going to work though and you might be left with a dull spot.

I'd be tempted to try sandpaper, and find some matching acrylic paint to reseal it.

Mr Clean magic eraser with some isopropyl alcohol, give it a scrub?

You can not.

This is not just a surface stain, but it has literally changed the material underneath the burn.

Amy attempted fix will just be an obscene amount of work for what will still be an obvious repair job.

Your nice list doesn't seem to have non polar solvents in case it's greasy. Something like soap, alcohol, (turpentine, acetone, xylene).

But yeah, might not be a surface stain as the other comments suggest.

Acrylic is plastic, you likely burned the plastic with hot ashes. You can try wet sanding it down using very fine grit sandpaper, followed by polishing, but YMMV.

rubbing alcohol might help but your probably going to have to buff it out with a Dremel