We establish an explicit rate of convergence for some systems of mean-field interacting diffusions with logistic binary branching, towards solutions of nonlinear evolution equations with non-local self-diffusion and logistic mass growth, which were shown to describe their large population limits in [12]. The proof relies on a novel coupling argument for binary branching diffusions based on optimal transport, allowing us to sharply mimic the trajectory of the interacting binary branching population by means of a system of independent particles with suitably distributed random space-time births. We are thus able to derive an optimal convergence rate, in the dual bounded-Lipschitz distance on finite measures, for the empirical measure of the population, from the convergence rate in 2-Wasserstein distance of empirical distributions of i.i.d. samples. Our approach and results extend propagation of chaos techniques and ideas, from kinetic models to stochastic systems of interacting branching populations, and appear to be new in this setting, even in the simple case of pure binary branching diffusions.