Some ideas: Keep an eye on Craigslist for people selling used boxes. A cargo basket & cargo net instead of a box. A car top soft storage bag instead of a box (View Post)
I haven't heard of HDPE warping in normal use, even in hot climates. https://www.tapplastics.com/product/plastics/cut_to_size_plastic/hdpe_cutting_boards/346 (View Post)