This paper presents a staged deep-learning framework for robust modeling of noisy multi-input multi-output (MIMO) Hammerstein systems under correlated measurement disturbances. The proposed architecture preserves the Hammerstein decomposition by combining a gated recurrent nonlinear block for latent nonlinear transformation, an ARX-type recursive linear dynamic block for deterministic output reconstruction, and a correlated residual-noise model with heavy-tailed innovations. To improve training stability and robustness, the learning process is organized into three stages: robust initialization from step-response data, deterministic recovery using the simulator-provided noise-free target in a synthetic benchmark setting, and joint fine-tuning under measured noisy outputs. The proposed method is validated on a synthetic 2-input/2-output benchmark with latent dimension two using a five-seed evaluation protocol. On the deterministic target, the framework achieves mean ± standard deviation test performance of RMSE 0.2168±0.0454, R2 = 0.9444 ± 0.0240, FIT = 76.99 ± 5.14%, and wMAPE = 10.89 ±1.55%. Under measured noisy outputs, it remains stable and competitive, yielding RMSE 0.2948 ± 0.0318, R2 = 0.9050 ±0.0233, FIT = 69.40 ± 3.76%, and wMAPE = 22.69 ± 1.06%. Comparisons against a direct recurrent baseline and a linear ARX baseline further show that the proposed framework consistently improves over both unstructured deep prediction and purely linear modeling, while remaining stable across random seeds. Overall, the results indicate that the proposed framework is best interpreted as a robust deep MIMO Hammerstein modeling strategy rather than an exact physical parameter identification scheme. The method provides a practical compromise between structural interpretability, recurrent nonlinear representation, and robustness to noisy measurements. MSC (Mathematics Subject Classification): 93B30; 93C10, 68T07, 93B40.