My only quick-reaction suggestion would be to use a different, thinner fabric on the underside of the placard that will hide the zipper. This might make the thickness more manageable. Is the zipper to be in the center front? or in a back or side seam? Is there any other closure option such as buttons, clasps, or snaps, that would work?
Good luck with the zipper. I bought the special zipper foot for the invisibles, but have never managed to make one look really good, and have stopped trying.
For what it’s worth, my sewing teacher back in Home Ec class said that the word for the little flap enclosure for a regular zipper is “placket.” Most of us have learned sewing by talking, looking and listening, not from reading, and it sounds almost like placard! (which is a cardboard poster.)