Is the scratch in the actual sapphire, or just in the AR coating? To me it almost looks like just in the AR coating, and if so, the 0.5 micron is all you need.
If the scratch is actually in the sapphire, then you and the next 3 generations of your family might be able to complete removing it by rubbing on the crystal with your finger. If you and your heirs don't mind doing this for the next several decades that's fine, but if you want to speed the process up, you will need some power tools.
This is the one, single, solitary application I have for the dreaded Dremel tool.** Use the small hard felt buff, spread some paste on the buff and a bit on the crystal, and buff away. As with any polishing process, be sure to remove all the scratches made by the previous abrasive before moving on or you will be starting over again. Make sure you use a separate buff for each grade of abrasive, and also clean the surface well between grades. Be prepared for this to take a couple of days to complete, and also be prepared to have some slight distortion in the crystal where the material was removed.
Those saying to take it to a professional are out to lunch IMO. Unless you have buckets of money to pay a professional to spend a couple of days polishing your crystal, this is most certainly a DIY job. If you are not comfortable doing a DIY job, take it to a professional yes, but not for polishing - to have the crystal replaced with a new one. Trust me the cost of a new crystal and the labour to install it is going to be multiple less than the labour someone like me will charge you to polish out those scratches. The only time I actually polish sapphire crystal is where I am unable to source a new replacement crystal.
** - The only reason I suggest using a Dremel instead of a proper polishing machine is that to charge a large diameter buffing wheel with diamond paste would cost a fortune, so a very small wheel on a hand held device is the only realistic option.
Cheers, Al
