Inverse design of shell-based metamaterials with multiple complex functionalities, such as high energy dissipation, prescribed nonlinear responses and localized deformation modes, remains challenging due to the vast design space and the high cost of accurate simulations, particularly when controlling both macroscopic and local field responses. Here, we present a data-efficient AI-driven framework for the multi-objective inverse design of shell metamaterials using graph neural networks and neural operator surrogates embedded within Bayesian optimization loops. Surrogate models, including MeshGraphNet-style architectures and point-cloud-based neural operators, are trained on only a few hundred high-fidelity finite element simulations to learn structure-property relationships. Unlike purely global surrogate models, the proposed approach explicitly predicts local quantities, such as stress and strain fields, in addition to macroscopic responses including the effective stiffness tensor and nonlinear compressive behavior. These local field predictions act as strong physical regularizers, enabling accurate global predictions in the low-data regime while allowing direct targeting of localized functional requirements. The trained surrogates are integrated into a multi-objective Bayesian optimization framework to guide the inverse design of multifunctional shell-based, triply periodic metamaterials. Compared to diffusion-based generative approaches, the proposed framework requires up to two orders of magnitude fewer training samples while achieving prediction errors below 10%, demonstrating substantially improved data efficiency. The framework is demonstrated through a real-world application to bone scaffold design, where both global stiffness matching and local stress distributions are optimized to mitigate stress shielding while preserving mechanical integrity and load transfer. Beyond biomedical scaffolds, the proposed framework is applicable to lightweight load-bearing structures, impact-absorbing shells, architected energy absorbers, and acoustic or elastic wave-controlling materials. By explicitly incorporating local field information into surrogate modeling and optimization, this approach enables efficient discovery of high-performance metamaterials in regimes where high-fidelity data are inherently scarce.