Mesh-based numerical methods, such as the finite element method formulated within a Lagrangian framework, encounter significant difficulties when modeling air flows and air-structure interaction (ASI) problems involving shock waves, primarily due to severe mesh distortion and entanglement. In contrast, meshfree methods such as the Reproducing Kernel Particle Method (RKPM) and the Material Point Method (MPM) offer notable advantages, as their particle-based descriptions naturally avoid singularities associated with mesh distortion. Nevertheless, these meshfree approaches still suffer from various numerical artifacts, and additional stabilization is essential when simulating compressible flows with shocks. To address these challenges, we propose a Reproducing Kernel Enhanced Material Point Method (RK-MPM) for the simulation of inviscid compressible air flows with ASI. A mixed formulation is adopted to solve the Lagrangian momentum and energy conservation equations, while the reproducing kernel (RK) approximation is employed to mitigate cell-crossing instabilities inherent to standard MPM formulations. To further suppress shock-induced oscillations, the classical artificial viscosity is enhanced through tensorial formulations of the diffusion and divergence operators, guided by the vorticity modulus. This strategy enables an adaptive blending of numerical dissipation between the dilatational and deviatoric components, leading to improved stabilization. In addition, a Hamiltonian-type Nitsche formulation is derived to impose air-structure interfacial conditions, in which displacement and velocity constraints are enforced through properly regularized point-to-node projections, effectively preventing numerical leakage. The proposed stabilized RK-MPM framework is validated using a series of benchmark problems, including shock wave propagation, vortical compressible flows, and representative air–structure interaction scenarios. Numerical results demonstrate enhanced stability, accuracy, and robustness compared with conventional MPM formulations, underscoring the method’s potential for complex compressible flow and ASI applications involving strong shocks.