IIRC you can remove the cover completely and paint it separately. I would do that rather than hassle with taping off the LEDs but sure, you can paint the cover.
Only recently has it included a cover. Mine is just some LEDs on a PCB.
I doubt there is anything on the PCB that would be damaged as long as the paint isn't conductive.
Andy would probably sell you a cover is you emailed him.
ArcadeGuns.com used to sell a cover (before Ultimarc had them), but it doesn't look like they sell separate parts anymore.