Auxetic metamaterials, characterized by a negative Poisson’s ratio (NPR), exhibit lateral expansion under tension and densification under compression. These deformation mechanisms are attractive for soft robotics and flexible mechanical systems, where large deformations are often required. With recent advances in additive manufacturing (AM) technologies, the fabrication of auxetic metamaterials with complex microstructures has become feasible. In practice, achieving such auxetic responses relies on hyperelastic materials under large deformations and involves strong geometric nonlinearity and structural instability, which pose substantial challenges for computational design. However, conventional gradient-based topology optimization (TO) methods are often inadequate due to numerical instability, difficulties in sensitivity evaluation, and high computational costs. To address the above challenge, we propose a data-driven topology design framework for computationally designing hyperelastic auxetic metamaterials with enhanced large-deformation capability and structural stability. The framework uses a multifidelity strategy, combining linear elastic TO for initial exploration with a high-fidelity model that incorporates hyperelastic constitutive behavior and self-contact phenomena for accurate large-deformation assessment. Elite design candidates are used to train a convolutional variational autoencoder to enable efficient exploration of the high-dimensional design space and to identify the architectures exhibiting enhanced NPR and improving buckling resistance. The high fidelity computational model captures nonlinear material behavior and contact but neglects manufacturing imperfections and additive manufacturing anisotropy. These factors may critically influence large-deformation responses. Therefore, experimental validation is conducted to assess the robustness and practical reliability of optimized designs. The selected configurations are fabricated using stereolithography (SLA) with a commercial hyperelastic resin (Elastic 50A V2) and subjected to quasi-static compression tests. The experimental results are compared with high-fidelity simulations to quantify deviations in NPR and structural stability, thereby bridging computational predictions and physical realization.