Mine was held on by upholstery screws right through the tub. It's by far the easiest and you can get colored screws or caps to clean up the look if you so choose. An alternative would be industrial velcro strips in some spots that you don't want hardware to show.
The plywood is 6mm, 3mm would work just as fine. It will have a thin layer of scrim foam and faux leather covering. All in probably about 8-9mm thick. You don't really lose any appreciable space with this approach and you benefit from the fact the panels can be removed easily by pulling out the Fir Tree clips which are a tight push fit.