This study presents a robust computational framework for the design and analysis of energy-dissipating mechanical metamaterials. The primary dissipation mechanism stems from the elastic buckling of periodic lattice microstructures, which are modeled using beam elements under a small-strain, large-displacement kinematic regime. By leveraging these geometric instabilities, the architected material can effectively attenuate vibrations and manage energy transmission from external excitations. To overcome the prohibitive computational overhead associated with multiscale simulations (FE2), we propose a data-driven homogenization strategy based on Gaussian Process (GP) surrogates. The model learns the complex constitutive mapping between the macroscopic strain tensor and the resulting Piola-Kirchhoff stress. A pivotal advantage of this GP-based approach is the ability to derive the algorithmic tangent stiffness matrix analytically from the surrogate. This enables the seamless integration of homogenized microstructural behavior into macro-scale finite element solvers, achieving high-fidelity results while maintaining computational tractability. The reliability of the surrogate is ensured through an active learning protocol by monitoring the uncertainty in the posterior distribution of the surrogate function. As a result, the training dataset is enriched in regions of the design space where the model's predictive confidence is lowest. This targeted sampling ensures high global accuracy with minimal data points. Finally, a comparative assessment demonstrates that for this class of nonlinear mechanical problems, Gaussian Processes provide a more robust and straightforward alternative to traditional artificial neural networks, particularly regarding uncertainty quantification and gradient consistency.