To address the issue of insufficient flexibility in integrated energy systems (IES) caused by output uncertainties on both supply and demand sides, this paper constructs a hydrogen-enriched compressed natural gas (HCNG)-based IES with deep gas-hydrogen coupling, and proposes a distributionally robust capacity optimization strategy considering a flexibility penalty mechanism and a tiered carbon trading mechanism. First, based on the characteristics of HCNG, a cooperative optimization framework for HCNG units, gas turbines, gas boilers, and natural gas pipelines is established to fully exploit hydrogen utilization potential. Second, to facilitate low-carbon system transition, a tiered carbon trading mechanism, featuring energy users and carbon operators as the main participants, is constructed within the carbon trading market to effectively constrain system carbon emissions. Finally, by quantifying source-load uncertainties using probability distributions, a flexibility supply-demand model is established, and a distributionally robust capacity optimization model is developed. Through scenario-based comparisons in an IES consisting of the IEEE 9-bus power system, a 7-node natural gas system, and a 6-node thermal system, the proposed model’s effectiveness in improving the low-carbon performance, economic efficiency, and flexibility of IES is validated.