How to secure screws in plywood going with grain??

Another thought: it might not look pretty, but maybe you could attach the plastic/insulation piece to the sides with Eternabond tape, with maybe a couple of inches on the side and a couple on the plastic, and then screw the trim in over the tape if you wanted. The idea is that the tape might provide the majority of the structural support for the skin. I honestly don't know if that would be practical or not, but it sounds potentially workable at the moment.