It is highly challenging to register large-scale, heterogeneous SAR and optical images, particularly across platforms, due to significant geometric, radiometric, and temporal differences, which most existing methods struggle to address. To overcome these challenges, we propose Grid-Reg, a grid-based multimodal registration framework comprising a domain-robust descriptor extraction network, Hybrid Siamese Correlation Metric Learning Network (HSCMLNet), and a grid-based solver (Grid-Solver) for transformation parameter estimation. In heterogeneous imagery with large modality gaps and geometric differences, obtaining accurate correspondences is inherently difficult. To robustly measure similarity between gridded patches, HSCMLNet integrates a hybrid Siamese module with a correlation metric learning module (CMLModule) based on equiangular unit basis vectors (EUBVs), together with a manifold consistency loss to promote modality-invariant, discriminative feature learning. The Grid-Solver estimates transformation parameters by minimizing a global grid matching loss through a progressive dual-loop search strategy to reliably find patch correspondences across entire images. Furthermore, we curate a challenging benchmark dataset for SAR-to-optical registration using real-world UAV MiniSAR data and Google Earth optical imagery. Extensive experiments demonstrate that our proposed approach achieves superior performance over state-of-the-art methods.