Hyperelastic constitutive models form the basis for modeling large deformations in elastic solids. Traditional approaches use a prescribed strain-energy function with material-specific parameters calibrated from experiments. While this approach is well suited for modeling individual materials, it is less effective for constructing a single parametric model across materials with varying composition, since limitations of the fixed constitutive form can introduce model-form mismatch beyond parameter calibration. In many emerging applications, including multi-material additive manufacturing, there is a growing need for constitutive models capable of accommodating large variations in material behavior. To address this challenge, recent work has explored flexible, data-driven representations of strain-energy functions that preserve key physical constraints. Widely used examples include input convex neural networks (ICNNs) and partially input convex neural networks (pICNNs), which support material stability and numerical robustness. Prior work has explored monolithic parametric ICNNs that incorporate material descriptors and composition variables as additional inputs to represent families of materials within a single model. However, the constrained optimization landscape imposed by convexity-preserving model weight restrictions can make neural network training challenging, especially for learning widely varying material responses and from ragged data ranges. We propose a data-driven modeling framework that can generalize across hyperelastic materials by using a shared low-dimensional convex basis in strain-energy space. This representation enables learning a wide range of material responses while accommodating heterogeneous datasets across materials. The shared low-dimensional representation improves optimization robustness and offers a scalable, interpretable framework for multi-material hyperelastic modeling and prediction of unseen materials.