This work proposes a three-dimensional geometrically nonlinear topology optimization framework for bi-modulus hyperelastic materials based on the Moving Morphable Components (MMC) method, aiming to investigate the influence of tension–compression asymmetry on the optimized structural design. Leveraging the decoupling between geometric description and finite element analysis, a redundant degree-of-freedom removal technique is adopted to avoid mesh distortion induced by low-stiffness regions in geometrically nonlinear topology optimization. To further alleviate the convergence difficulties arising from the non-smooth constitutive, a constitutive-parameter-based homotopy continuation (CPHC) strategy is applied, providing a smooth transition of material parameters and significantly improving the robustness of Newton–Raphson iterations. Numerical examples demonstrate that the optimized design under finite deformation strongly dependent on the tension to compression modulus ratios and quite different from the one obtained within the assumption of classical hyperelastic materials. The results confirm the effectiveness and numerical stability of the proposed computational framework in the presence of combined geometric nonlinearity and material asymmetry, and further highlight the engineering necessity of tension–compression asymmetric material behavior in structural topology optimization design.