If you're using the Frogger-style cab, then one hurdle is going to be working around the lip of the control panel. You'll probably have to have the plexi in two pieces (one piece behind CP, then the door itself).
I'd definitely recommend trying to get it to swing sideways or up. If it swings down, you're going to have to reach over 18" or so of plexi to reach the monitor. Also, It's probably going to make contact with the control panel and probably get scratched up.
I think you might be onto something with the black bezel, but I'm thinking black advesive vinyl (used by sign shops) might work better and be more of a permanent solution instead of a loose piece of cardboard.
Another issue is that you will have to leave a little area on the edge of the plexi for either a hinge mounted on the very edge, or clear out some area behind the plexi so the trailing edge (and thickness) of the plexi has somewhere to go. Maybe you could make a piano hinge not THAT obvious if it's on the very top edge
BTW, my MAME cab was a knock-off of the Sega cabs.