How do you remove rust stains from bathroom sink?
I'd stick with CLR or vinegar : use plumber's putty to make a ring and make a small pool.  Hydrogen peroxide would be another choice (not concurrently, with ANY of the other choices)

Its location suggests a leaky soap dispenser.

Solid surface materials can be sanded (starting at 220) and then up through the grits (320, 400, 600), finishing with auto body rubbing compound then polishing compound.