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Decarbonizing the energy system requires large-scale integration of renewable energy, but its variability presents challenges for reliability and long-duration storage. Power-to-gas technologies present a promising solution by converting surplus electricity into gaseous fuels. Nonetheless, current approaches often overlook the carbon cycle in the system and the potential of natural gas infrastructure as a storage medium. This study proposes and evaluates an integrated renewable power-to-gas and hybrid gas grid system that retrofits pipelines to enable the co-injection of CH4 and CO2, where a dynamic carbon cycle is established using membrane-based gas separation. The system is modeled through a mixed-integer linear programming (MILP) optimization framework combined with a quasi-dynamic gas grid model, capturing both operational and compositional effects. The model undergoes a year-long simulation based on varying electricity and gas demand profiles and market information. Results indicate that the hybrid gas grid is technically feasible and economically competitive, particularly for long-distance transmission and seasonal storage. An optimal CO2 concentration of 30 % in the gas grid minimizes the levelized cost of methane (LCOM) to 0.378 EUR/kg. The findings provide valuable insights for retrofitting gas grids to support renewable integration, carbon cycle, and sector coupling under future low-carbon energy scenarios.