One emerging research area within the fields of acoustic and elastic metamaterials involves designing subwavelength structures that display elastic instabilities in order to generate an effective medium response that is strongly nonlinear. To capture the overall frequency-dependent and dispersive macroscopic response of such heterogeneous media with subwavelength heterogeneities, a theoretical framework is developed that accounts for higher-order stiffnesses of a resonant, nonlinear inclusion that varies with a macroscopic pre-strain, and the inherent inertia associated with an inclusion embedded in a nearly incompressible elastic matrix material. Such a model can be used to study varying macroscopic material properties as a function of both frequency and pre-strain and the activation of such microscale instabilities due to an external, macroscopic loading, as demonstrated with a buckling metamaterial inclusion that is of interest due to its tunable and tailorable nature. The dynamic results obtained are consistent with similar static behavior reported in the literature for structures with elastic instabilities.