Topology optimization of hyperelastic structures undergoing large deformations remains a challenging task, particularly for rubber-like materials described by the Ogden model [1]. Although advanced discretization methods such as the virtual element method have been proposed to mitigate mesh distortion in such materials [2], standard finite element solvers frequently fail in high-compression regimes due to severe volumetric locking and element inversion. Furthermore, low-density elements in the void domain often exhibit excessive distortion, leading to negative Jacobians and indefinite stiffness matrices, which inevitably halt the optimization process [3, 4]. This study presents a robust topology optimization framework specifically designed to withstand severe deformations. We integrate three key stabilization techniques to address these numerical instabilities comprehensively. First, we employ a Selective Reduced Integration (SRI) formulation that decouples the strain energy into volumetric and deviatoric terms, effectively resolving volumetric locking without the computational cost of mixed finite element methods. Second, we introduce an energy interpolation scheme [4] extended to the Ogden model, which smoothly transitions the constitutive law of low-density elements from hyperelasticity to a stable linear elastic model, thereby preventing numerical singularities. Third, we implement an adaptive nonlinear solver equipped with a geometrically constrained line search and a best-solution preservation strategy, enabling the solver to recover safely from potential divergence. Crucially, we derive exact analytical sensitivities that are strictly consistent with both the SRI formulation and the energy interpolation scheme, ensuring mathematical rigor and convergence stability. Numerical examples, focusing on large-deformation problems such as a cantilever beam, demonstrate that the proposed framework successfully generates optimal designs even under severe geometrical nonlinearities. The results highlight the method's capability to handle mesh distortion issues in regimes where conventional optimization approaches typically diverge.