As Pats300zx said above try something a bit milder. If that doesn't work, try some Brasso.
We (Old Army JROTC) used to use Brasso on out name tags which is a white sheet of plastic with a thin sheet of black plastic over it. The black sheet was pebbled in texture and we used a scotch brite pad to smooth it out, then spent a good day rubbing Brasso into the plastic and oh boy the resulting finish was smooth and glossy black. Now I don't have a nametag to wear on my uniform.
This is different than clear plastic though. It will work, but you have to follow up with something finer.