Internally resonant mechanical systems exhibit complex yet intriguing nonlinear dynamics that can be harnessed in the design of next-generation MEMS devices and energy harvesters. While shape and topology optimization offer great potential for tuning these nonlinear vibrations, they face the curse of dimensionality in gradient-based optimization schemes. To address this challenge, we employ spectral submanifold (SSM) reduction theory, which transforms high-dimensional, internally resonant systems into low-dimensional reduced-order models (ROMs). Using these SSM-based ROMs, we formulate optimization problems and develop efficient algorithms for computing nonlinear vibration responses and their design sensitivities. The effectiveness of the proposed framework is demonstrated through the design of nonlinear MEMS devices, achieving targeted performance optimization.