If it were me...
and it was me, once, but I wasn't able to advise myself from experience - I'd smooth the edges of the deep scratch and then leave it alone.
I tried sanding out a deep scratch by approved methods and ended up with a rather largish flat spot where the scratch used to be, quite a bit of optical distortion, and later the whole area crazed. I made one effort to polish out the crazing which didn't help things at all. Wish I had left it alone, because the repair was worse than the scratch ever was. At least it's in the passenger's line of sight, not mine.
My counsel for a deep scratch: smooth the edges, and step away.
-Stormy