In the context of "carbon peak" and "carbon neutrality," the increasing adoption of carbon capture technology and hydrogen energy utilization is expected to pose significant challenges to complicate the interactive dispatch of multi-microgrid systems. This study develops a microgrid system model integrating calcium-based carbon capture with hydrogen energy utilization, demonstrating that combined with hydrogen systems, the thermochemical energy storage properties of calcium-based materials can significantly reduce both dispatch operation costs and carbon emissions in microgrid systems. Based on Nash bargaining theory, we establish a cooperative operation model for multi-microgrid power sharing, employing the alternating direction method of multipliers to obtain Nash equilibrium solutions for subproblems. For the cooperative benefit allocation subproblem, negotiation is conducted through an asymmetric Nash bargaining approach. The case study results demonstrate that combined with power-to-gas system, through energy sharing among microgrids, the calcium-based carbon capture enhances the overall alliance operational efficiency by approximately 4.33% and reduces carbon emissions by 4.09%. The asymmetric bargaining model based on contribution values ensures that microgrids with greater energy contributions receive more benefits, achieving a fair distribution of cooperative gains.