Edgewise compression response of a composite structural insulated panel (CSIP) with magnesium oxide board facings was investigated. The discussed CSIP is a novel multifunctional sandwich panel introduced to the housing industry as a part of the wall, floor, and roof assemblies. The study aims to propose a computational tool for reliable prediction of failure modes of CSIPs subjected to concentric and eccentric axial loads. An advanced numerical model was proposed that includes geometrical and material nonlinearity as well as incorporates the material bimodularity effect to achieve accurate and versatile failure mode prediction capability. Laboratory tests on small-scale CSIP samples of three different slenderness ratios and full-scale panels loaded with three different eccentricity values were carried out, and the test data were compared with numerical results for validation. The finite element (FE) model successfully captured CSIP’s inelastic response in uniaxial compression and when flexural action was introduced by eccentric loads or buckling and predicted all failure modes correctly. The comprehensive validation showed that the proposed approach could be considered a robust and versatile aid in CSIP design.