This study addresses the system identification of a small autonomous surface vehicle (ASV) under moored conditions using Hankel dynamic mode decomposition with control (HDMDc) and its Bayesian extension (BHDMDc). Experiments were carried out on a Codevintec CK-14e ASV in the towing tank of CNR-INM, under both irregular and regular head-sea wave conditions. The ASV under investigation features a recessed moon pool, which induces nonlinear responses due to sloshing, thereby increasing the modelling challenge. Data-driven reduced-order models were built from measurements of vessel motions and mooring loads. The HDMDc framework provided accurate deterministic predictions of vessel dynamics, while the Bayesian formulation enabled uncertainty-aware characterization of the model response by accounting for variability in hyperparameter selection. Validation against experimental data demonstrated that both HDMDc and BHDMDc can predict the vessel's response to unseen regular and irregular wave excitations. In conclusion, the study shows that HDMDc-based ROMs are a viable data-driven alternative for system identification, demonstrating for the first time their generalization capability for a sea condition different from the training set, achieving high accuracy in reproducing vessel dynamics.