Well yeah, my initial thought on what material to use while reading your post was Perspex/Plexiglass, so I'd go with another sheet of that.
The best way to cut the whole without cracking the plastic, in my opinion would be to use an electric drill with either a masonry chisel
bit or a holesaw
bit. This will give you the most accurate and smooth results. Then once you've cut the hole, I'd recommend rounding the edges off with some fine emery cloth.
I hope this helps. If you have any problems, just reply to this post or PM me.