I have used screws to secure the plexi. And I have secured at top and bottom by being behind the speaker panel and control panel, and use nothing on the sides. But I don't think either would be a good option on your project.
Can you not lower the monitor a bit? Maybe then you would have room to put L shaped aluminum, turned so one side goes between monitor and wood panel, that way it looks like a flat piece going down each side of the plexi.
Im sure it's too late to do it, but it would be nice if you could route a slot on each side to allow the plexi to slide into the wood, the edge of the plexi would then be completely hid.