This paper presents a cooperative control strategy to simultaneously address energy balancing and circulating current suppression in arm multiplexing modular multilevel converter (AM-MMC). While AM-MMC reduce hardware costs and improve submodule utilization by arm multiplexing, they introduce challenges such as energy coupling and amplified circulating currents, particularly the dominant second-order harmonic component. The proposed solution integrates a hierarchical energy control architecture with a proportional-resonant (PR) controller. First, a mathematical model of the AM-MMC is developed to analyze its energy interaction mechanisms and circulating current spectral characteristics. A fast sorting-based capacitor voltage balancing algorithm is employed to ensure rapid energy equilibrium among bridge arms. Additionally, the PR controller replaces conventional PI control, offering high-gain precision at 100 Hz to effectively suppress second-order harmonic circulating currents. Simulation results verify that the combined strategy limits circulating currents to within 15% of the rated value and restricts capacitor voltage fluctuations below 10% under both steady-state and transient conditions. These improvements enhance the operational reliability and efficiency of AM-MMC, making them more viable for high-voltage direct current (HVDC) transmission systems.