In this paper, we propose a novel transmissive reconfigurable intelligent surface (TRIS) transceiver-empowered distributed cooperative integrated sensing and communication (ISAC) network to enhance coverage as well as to enhance wireless environment understanding. Based on the network requirements, the users are categorized into cooperative users (CUEs) and destination users (DUEs), and the CUEs utilize their own resources to serve the DUEs. To realize cooperation, we implement rate-splitting multiple access (RSMA) at the base station (BS), where the common stream is decoded and reencoded at the CUEs and forwarded to the DUEs, while the private stream satisfies the CUEs' own communication requirements. We construct an optimization problem with maximum minimum radar mutual information (RMI) as the objective function to optimize the BS beamforming matrix, the CUE beamforming matrices, the common stream rate vectors, and the user scheduling vectors. Due to the coupling of the optimization variables and non-convex operation, the proposed problem is a non-convex optimization problem that cannot be solved directly. To address the above challenges, we adopt a consensus alternating direction method of multipliers (ADMM) framework to decouple the optimization variables and solve it. Specifically, the problem is decoupled into multiple subproblems and solved by iterative optimization independently until overall convergence is achieved. Finally, numerical results validate the superiority of the proposed scheme in terms of improving communication sum-rate and RMI, and greatly reduce the algorithm complexity.