Although the flexible origami gripper can handle a wide range of objects, there is a need for significant further improvement in its gripping performance. This study develops a novel nonlinear topology optimization (NTO) method to enhance the gripping performance of an origami chomper-based flexible gripper. The proposed NTO method incorporates the additive hyperelasticity technique and multi-resolution design (MRD) strategy with the advantages of being computationally efficient, having excellent convergence, and enabling refined design. The effectiveness of the proposed NTO method is validated by two compliant mechanism benchmark examples, i.e., the displacement inverter and gripper mechanisms. We apply the NTO method to the origami chomper-based flexible gripper to redistribute the material at the creases to obtain the optimized origami chomper-based flexible gripper. Several optimized origami chomper-based flexible gripper prototypes are fabricated by using laser cutter, followed by a series of experiments to test the gripping performances, including gripping range capability under an identical input load, maximum gripping ratio, gripping adaptability, and achieving richer gripping characteristics by size scaling. Results demonstrate that the optimized origami chomper-based flexible gripper can handle a wide range of objects irregularities in textures and uneven shapes;and the gripping range capability can be significantly enhanced by the NTO method. We also show that the optimized origami chomper-based flexible gripper can enable effective gripping of objects across scales from millimeters to centimeters to decimeters through size scaling.