If you put gloss laminate under lexan, you're going to get parts of the laminate sticking to the lexan. Particularly, anyplace where the surface of the panel is a tiny fraction higher, it's going to push it up to touch the lexan. Any place where you mount buttons, they're going to put some pressure on the lexan -- probably enough to push it down a tiny fraction.
How much moisture are you going to get under the lexan?
I would consider simply sealling the paper with clear acrylic spray (aka spraypaint). Maybe waterproof paints, if you really think there's going to be a problem.
Alternately, with thick laminate, you can eliminate the lexan entirely. Just rubber cement the overlay and mount the controls through it. You probably wouldn't get away with that in a real arcade, but hopefully no one's going to peel up your edges in your house ;-)
Bob