Please consider that you will likely end up with little bits of polystyrene all over your section over time. Some absolute specimen has at some point dumped some EPS and bean-bag balls up the back of my property, now it is all through the soil 50+m from where it was dumped. I have spent hours digging through the soil trying to remove it, but only covered an area of 1sq.m, 300mm deep - horrible shit. Could you stick to your original plan of making a chiller, and find an alternative material for the shower?
What's the brief for the shower? Does it have to be portable? I have some friends who just made a shoulder-height, rectangular enclosure, with a door opening on to a bit of decking for a changing area, next to a shower tray. They lined the inside with some leftover colour steel (or equiv.), the outside with thin ply, and had a hose running through a gas califont feeding a shower head. Easy, and you get to look at the stars while you're showering... also no polystyrene floating round your section.
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