I might be able to help here as I have significant experience in polishing plastic and paint. I would suggest first cleaning the screen with a clean microfiber towel and some mild cleaner (windex would probably be best as the ammonia or alcohol will remove any waxes/glazes/silicone from the screen so you can see exactly where you are starting). Next you want to start with the most mild polish you can find and lightly polish the screen with another clean microfiber towel. It is important to always start with the LEAST abrasive polish possible - you can always polish it again if it isn't abrasive enough. Using a more aggressive polish off the bat will more than likely lead to more polishing later. A high quality automotive wax might just do the trick for you here. Rub some of the wax after vigorous shaking to mix between your fingers and check the level of abrasive and if it is really fine it should work just fine for this use.