HDPE works OK. My only objection is the weight. High Density is the correct name. Two carbon arrow shafts (at around 1oz) would weigh considerably less than the ~6-8oz the HDPE weighs and provide more stiffness in the vertical 18" or so that it would cover. Your sleeping pad will also provide additional firmness if you use the 1/4" pads commonly available. Or just lay it flat against your back if it is too large to fit.
I did much the same with my old Trek. I removed the 1/4" foam pad and sewed the bottom closed. I added a second piece of fabric around the upper and lower corners after slitting the rear open. I slipped in two arrow shafts on either side and stitched along them, pretty much making them permanent (though the foot would only get within about an 1/8" of the shafts.) This enabled me to carry about 30-40 pounds comfortably with no pack collapse. Once I added my pad (1/2"x20x10 in 5 layers,) this was really comfortable.
Alternatively, you can add some external pad keepers (1/3 length pockets made from some mesh) that lets you add your pad to the pack. A 5 layer, 11" wide sleeping pad will work for warmer months with your pack under your feet and a stuff sack for a pillow. 5 payers will support about 25-30 pounds depending on weather you use dimpled pads, plain pads or inflatables. Just check to make sure your hip belt still works since it does change with/without the pad. You can use the existing pocket for other things.