Past high-energy density laboratory experiments provided insights into the physics of supernovae, supernova remnants, and the destruction of interstellar clouds. In a typical experimental setting, a laser-driven planar blast wave interacts with a compositionally-homogeneous spherical or cylindrical target. In this work we propose a new laboratory platform that accounts for curvature of the impacting shock and density stratification of the target. Both characteristics reflect the conditions expected to exist shortly after a supernova explosion in a close binary system. We provide details of a proposed experimental design (laser drive, target configuration, diagnostic system), optimized to capture the key properties of recent ejecta-companion interaction models. Good qualitative agreement found between our experimental models and their astrophysical counterparts highlights strong potential of the proposed design to probe details of the ejecta-companion interaction for broad classes of objects by means of high energy density laboratory experiments.