Back when I was shooting competitively, we used closed-cell-foam sleeping-bag pads.
There's a kind of closed-cell foam that's blue in color and only about 1/2" thick, but it's lightweight, it has a really tough skin, and it's really hard to damage. It used to be available cut-to-order, so you could get almost any size pad.
IPSC prone shooting used to start from standing, so the pad had to absorb not only a full-grown adult falling on it, but also abrasion from skidding against dirt and stones. This blue stuff really worked.