OK, since you sound determined to do something about this (please be careful

) I'll try to point you in the right direction. But [insert repeat of warnings here].
Go with 3K grit, not 2K. Yeah, it'll make a huge difference. Get the Meguiar's Unigrit stuff, don't use the "regular" stuff like from 3M. Soak it in a water/wash shampoo mix. Use a sanding block.
Do only *ONE LIGHT PASS* . Wipe and inspect. Repeat but don't keep going until you think it's perfect, by that time you'll have done too much. Just try to make it better; you can always do more at a later date but once you've gone too far you're SOL.
Remove the sanding scratches with a pretty aggressive compound that'll still work by hand/PC. I use the old 3M PI-III RC (05933) but it's getting hard to find these days.
Let us know how it goes and please take my warnings seriously.