This work deals with the design and validation of a control strategy for hybrid balancing systems (HBSs), an emerging concept that joins battery equalization and hybridization with supercapacitors (SCs) in the same system. To control this system, we propose a two-layer model predictive control (MPC) framework. The first layer determines the optimal state-of-charge (SoC) reference for the SCs considering long load forecasts and simple pack-level battery models. The second MPC layer tracks this reference and performs charge and temperature equalization, employing more complex module-level battery models and short load forecasts. This division of control tasks into two layers, running at different time scales and model complexities, enables us to reduce computational effort with a small loss of control performance. Experimental validation in a small-scale laboratory prototype demonstrates the effectiveness of the proposed approach in reducing charge, temperature, and stress in the battery pack.