Just to piss you off a little more... there used to be a clear plastic cover to protect the LCD screen but BMW removed it to save some bucks.
I doubt that you can polish that scratch as the surface is not smooth, as you can notice. An used screen may cost you some $500; brand new is about 3 times that.
-2015 F30 320i, Imperial/Oyster, 8AT: August 2014 ED
-2012 E92 M3 Jerez Black/Rust Brown, M-DCT: Gone
-2008 E90 M3 Melbourne Red/Bamboo, 6MT: Gone
-2007 MK5 GTI United Grey/Grey, DSG: Gone
-2004 E46 M3 Carbon Black/Cinnamon, 6MT: Gone