Human coordination often benefits from executing actions in a correlated manner, leading to improved cooperation. This concept holds potential for enhancing cooperative multi-agent reinforcement learning (MARL). Despite this, recent advances in MARL predominantly focus on decentralized execution, which favors scalability by avoiding action correlation among agents. A recent study introduced a Bayesian network to incorporate correlations between agents' action selections within their joint policy, demonstrating global convergence to Nash equilibria under a tabular softmax policy parameterization in cooperative Markov games. In this work, we extend these theoretical results by proving the convergence rate of the Bayesian network joint policy with log-barrier regularization.