A composite multifunctional radar absorption polymethacrylimide (RAPMI) foam sandwich (MRAPS) was constructed. Analytical models for out-of-plane and in-plane compression of the MRAPS were established. Three-dimensional failure mechanism maps and specific strength cubic cloud maps were drawn. Finally, a multiobjective particle swarm optimization (MOPSO) algorithm was used to achieve an integrated design, and a specific calculation example was tested and verified through experiments. The resulting sandwich with a 43.0 mm thickness core and 0.13 g/cm3 density achieves 90% effective absorption above broadband (RL ≤ −10 dB) in the 2–18 GHz range. As a radar absorption absorber, the MRAPS increased the engineering value and strategic importance of PMI foam for structural components in the military and aerospace sectors.