Better authorities on the subject might be where the home-theater mavens reside:
Ultimately, polycarbonate remains a relatively soft plastic as I’ve witnessed back in my RC car days (used as body shells) Once the protective sheet was pulled away, it would scratch/swirl if you looked at it funny.
There’s a whole spectrum that falls under “polishing”. If you’re using 3M polishing compound or toothpaste, that still may be too gritty (swirl-inducing) for clear plastics… and especially if just buffing by hand.
Not sure about the relative hardness of polycarbonate versus acrylic, but I recall in junior high shop class a century ago that the final stage for a wet-sanded, hazy block of acrylic was to then go over to a very fluffy-soft buffing wheel… can’t recall if it got dressed with anything (polishing wax, rouge), but it transformed the surface from a frosted haze to an optically clear one.
Only way to test if this applies would be to try it on a disposable DVD (ie, any Pauly Shore movie) first.
Another realm to look into comes from time spent with other roadster owners on cleaning the rear plastic window of the convertible top. Although these are vinyl windows, some of the products can/should apply to other plastics. One camp recommends spraying and polishing with Pledge. Others recommend Meguiar’s #17 & #10 and still others swear by Plexus Plastic Cleaner thru boating supply shops. Meguiar’s did the trick for me so I never explored anything else. Again, in the case of polycarbonate for a DVD, I’d still lean toward a machine buffing procedure to smooth away light scratches.