Hyperelastic shells exhibit significant potential for applications in cutting-edge fields such as soft robotics, metamaterials, and aerospace equipment, owing to their lightweight characteristics and high load-bearing capacity. However, the pronounced tendency of shell structures to undergo nonlinear large deformations poses substantial challenges for structural design, even with the aid of topology optimization (TO). In particular, topology optimization based on traditional low-order finite element methods often fails to meet the geometric and numerical continuity requirements inherent to shell simulations. To address these challenges, this paper proposes an Isogeometric Topology Optimization (ITO) framework specifically tailored for hyperelastic shells. NURBS basis functions used in Isogeometric Analysis (IGA) enables an accurate representation of shell geometry smoothness and deformation nonlinearity. To mitigate mesh distortion and improve the robustness of topology optimization, a line-search-based proxy mesh method is developed. By computing the critical step sizes that would cause element inversion or excessive stretching in each triangular element of the proxy mesh, the line-search step length for control-point updates is determined at each nonlinear iteration. The effectiveness of the proposed approach is validated using standard benchmark examples of the nonlinear ITO. Its robustness is demonstrated through the optimization of a classic paraboloidal shell. Numerical results confirm that the method maintains the stability of topology optimization under large deformation. The proxy-mesh method effectively suppresses element distortion, thereby significantly enhancing the robustness of the optimization process.