Recent results in the literature highlight the impact of nonlinear inertial forces on the post-flutter limit cycle oscillation (LCO) characteristics of highly deflected structures in supersonic axial flow. The current investigation examines how the ability to passively modulate nonlinear inertial forces may alter the overall aeroelastic response. The structural model is a one-dimensional nonlinear inextensible plate subject to nonlinear aerodynamic forces in accordance with a new, geometrically modified third-order Piston Theory. For the linear aeroelastic case, we find that nonhomogeneous mass distribution elicits discontinuous increases in the critical Mach number for flutter and several flutter mode-switching phenomena that are not observed when mass is added homogeneously. The existence of several different flutter mode mechanisms as a function of a concentrated mass location leads to different post-flutter LCO amplitude behavior. This is found to transition the underlying nonlinear structural dynamics to either be stiffening (when lower-order modes merge) or softening (when higher-order modes merge), which in turn alter the influence of nonlinear aerodynamic forces. We also address discrepancies in LCO amplitude trends due to the nonlinear inertial forces previously reported in the literature.