In some of my new laser cut kits we are using 2 pieces of light 3/16 , laminated together, to make the 3/8 instead of using a single 3/8 in piece, this makes for a very stiffer flap. I have a flat glass piece that keeps what ever I am gluing flat and straight while the glue cures, For items like this, i use a very thin layer of SLOW cure epoxy. I have not tried Gorilla type glues, but they may work well too
Randy