That's a good suggestion. My drawing board turned from a few scribbles on some graph paper to a convoluted mess of dozens of far-fetched ideas. I found that the piece I have can be peeled into five or six separate layers, each about 1mm in thickness, so I have a lot more design flexibility than I first thought. A good deal of research has gone into the design of lightweight, flexible insulation blankets for use around underwater pipelines where compression by hydrostatic pressure is a problem. That application is a good model for the challenges in sleeping pad design, I think. There are plenty of articles available that discuss underwater pipeline insulation blankets made of aerogel, EVA foam, urethane foam, syntactic silicone foam (filled with glass microballoons), and other materials.
EVA has low density and is waterproof and tolerates abrasion well. The aerogel blanket is dense, fragile, expensive (if you didn't obtain it for free), and complicated to use. For simplicity, durability, and price it can't beat EVA. The aerogel is a much better insulator than EVA per unit volume, but thinness isn't a requirement (or desirable) for sleeping pads. It could achieve better insulation per unit weight, but only in evacuated (less than 10 torr) envelopes. Without a vacuum the aerogel is hardly better than EVA per unit weight. I thought about using a small vacuum pump and a heat sealer to make thin, flexible aerogel vacuum panels. They would be an excellent insulator (almost as good as hard vacuum, like a dewar), but they would require valves for occasional re-evacuation, they would be fragile, and it's an expensive project.
So, I haven't decided yet how to use this stuff. Your sandwich idea is a good one. Other ideas are welcome. I'll post any new developments.