This paper addresses the energy balancing challenge in Arm Multiplexing Modular Multilevel Converters (AM-MMC) induced by circulating currents. While the AM-MMC reduces submodule count by 25% via time-division multiplexing, its operation generates second-harmonic circulating currents. These currents aggravate capacitor voltage fluctuations, disrupt arm energy balance, and increase losses. Theoretical analysis identifies a constant power component within the arm as the primary cause of energy imbalance. To solve this, an enhanced energy balancing method based on circulating current phase compensation is proposed. By controlling the initial phase of the injected compensating circulating current, this method eliminates the constant power component, significantly relaxing energy balancing constraints and enabling effective suppression of the circulating current amplitude. Simulation results demonstrate stabilized capacitor voltages, and improved output power quality. Compared to conventional approaches, the proposed strategy offers simpler implementation and faster dynamic response, providing a promising lightweight solution for high-reliability applications like offshore wind integration.