We study the propagation of chaos in a class of moderately interacting particle systems for the approximation of singular kinetic McKean-Vlasov SDEs driven by alpha-stable processes. Diffusion parts include Brownian (alpha=2) and pure-jump (1<\alpha<2) perturbations and interaction kernels are considered in a non-smooth anisotropic Besov space. Using Duhamel formula, sharp density estimates (recently issued in Hao, Rockner and Zhang 2023), and suitable martingale functional inequalities, we obtain direct estimates on the convergence rate between the empirical measure of the particle systems toward the McKean-Vlasov distribution. These estimates further lead to quantitative propagation of chaos results in the weak and strong sense.
