I don't like the construction adhesive idea because sooner or later that
sign is coming back down. It'll stick too well to the wall and not well
enough to the plastic. Maybe sooner than later when you realize it's
1/16" out of square.

I put up a bathroom mirror with the LN specified for mirrors and I think I
couldn't pull that off without some damage...

How about some furring strips lagged to the wall,
with the sign screwed to them?

My first thought. But he nixed that because the furring or screws may be
visible (it's about 10 ft from the ground looking straight up from the front
door).

Yes he's particular. That's why he's called the boss...

Sparky