To address the conflicts of interests among multiple stakeholders in regional integrated energy systems, as well as the issues such as high investment costs, uneven capacity utilization, and significant carbon emissions associated with user-side distributed energy storage, we proposed a low-carbon operation strategy for regional integrated energy systems based on three-level game among cloud energy storage service providers, integrated energy system operator (IESO), and load aggregators (LA). Firstly, an energy trading framework was established between the IESO and LA for leasing cloud energy storage. Secondly, considering the profit maximization demands of multiple rational stakeholders, a three-layer game model for the integrated energy system was established. The first layer is a principal-agent game with IESO as the leader and LA alliance as the follower; the second layer is a master-slave game with cloud energy storage service provider as the supplier and IESO as the receiver; the third layer is a cooperative game among LA alliance members, and the revenue is distributed using the asymmetric Nash bargaining method. Finally, the model was solved using the bisection method, KKT conditions, and the alternating direction multiplier method (ADMM). The simulation results show that the proposed strategy not only promotes the system's low-carbon operation, but also satisfies the economic needs of all stakeholders.