We simulate a system of coupled kicked tops to generate random density matrices distributed according to the Bures-Hall measure, which has an important role in quantum information theory. We study the effects of stochasticity and coupling parameters of the coupled-kicked-top system on the behavior of the associated bipartite pure state, eigenvalues and eigenvectors of the reduced state, and average entropies. For strongly chaotic phase space and adequate coupling between the constituting tops (subsystems), we find that the results of simulation agree with analytical results of random matrix theory. We also examine local fluctuation properties of the Bures-Hall eigenvalues using the distribution of nearest-neighbor spacing ratios. In the limit of high-dimensional density matrices, the empirical ratio distribution is found to approach the Wigner-surmise-like result for the Gaussian unitary ensemble. Additionally, we present some closed-form results for the Bures-Hall spectral density and corresponding moments using Meijer G functions. © 2021 American Physical Society.