Direct-drive wave power generation systems based on permanent magnet linear generators (PMLGs) produce fluctuating electromagnetic power under irregular wave excitation, which may affect DC-bus voltage stability and load-side power quality. To smooth the fluctuating output power, this paper develops an empirical mode decomposition (EMD)-based power allocation strategy for a battery–supercapacitor hybrid energy storage system (HESS). In the proposed strategy, EMD is used to decompose the fluctuating electromagnetic power into low-frequency and high-frequency components according to their time-scale characteristics. The low-frequency component is assigned to the battery for energy buffering, while the high-frequency component is assigned to the supercapacitor for transient power compensation. Finite-control-set model predictive current control (FCS-MPCC) is adopted on the generator side to improve the current response of the PMLG, and an MPC-based HESS controller is designed to track the assigned power commands and regulate the DC-bus voltage. Simulation results show a battery power-tracking error of 3.93 W and a DC-bus voltage standard deviation of 0.108 V; compared with LPF, EMD reduced the load-step voltage deviation by 11.94%. Experiments confirm that the PMLG back-EMF follows the translator velocity, the storage currents track their references, and the DC-bus voltage remains within ±2 V of its reference.