Well using Velcro would still require a Velcro strip be glued to the LCD.
I myself am in a similar situation, I have a 19" LCD that I want to use for the A-10c MFD information, and I want to attach the cougar panels without damaging the screen. I remember this rubber cement like glue I used when I was a kid called mucilage that was non toxic and was easily removed from non porous surfaces. I'm thinking this may be the way to go.