I've used Jiff, because it's cheaper than tooth paste, because I'm a cheap bastard, for plastic headlight covers, etc. Probably in the U S it's a different name, tooth paste like consistency for cleaning shower bases etc. That's for acrylic. I've got rid of scratches on polycarb with a soldering torch, about four inches from the screen looking at an angle that reflects, as soon as you see it go glossy, move to the next part. It wood be good to practice on the lower part, which isn't so important.
You need a lot of patience with heat, a little bit too long in one spot, it overheats and goes white. Even then you can file it out, but again you then need a lot of patience.