Try a nail buffing stick.
As these usually come with 2 or 3 different grades of abrasive, start with the coursest first, and as the scratches become less noticeable, switch to the next finer grade.
It's just like MikeV said, just all on a stick!
I find I can polish the plastic back to a perfect shine using these, and they are way cheaper than the sanding sticks most hobby shops sell.
I don't have a coffe filter at home, would instant work?
Karl