To obtain numerical estimates for the properties of a model for polymers in dilute theta solutions in its long-chain limit we follow a stochastic approach to polymer kinetic theory. The model takes into account configuration-dependent hydrodynamic interaction (HI) and simplifies to the Zimm bead-spring chain model in the case of preaveraged HI, for which parameter-free "universal ratios" such as the ratio between radius of gyration and hydrodynamic radius are known. The Chebyshev polynomial method and a variance reduction simulation technique are used to implement an efficient Brownian dynamics simulation. We resolve the full dependence of several characteristic ratios versus both chain length and hydrodynamic interaction parameter, we extrapolate their values to determine universal behaviors, and compare with analytical and experimental results.