The Dean-Kawasaki equation - one of the most fundamental SPDEs of fluctuating hydrodynamics - has been proposed as a model for density fluctuations in weakly interacting particle systems. In its original form it is highly singular and fails to be renormalizable even by approaches such as regularity structures and paracontrolled distributions, hindering mathematical approaches to its rigorous justification. It has been understood recently that it is natural to introduce a suitable regularization, e.g., by applying a formal spatial discretization or by truncating high-frequency noise. In the present work, we prove that a regularization in form of a formal discretization of the Dean-Kawasaki equation indeed accurately describes density fluctuations in systems of weakly interacting diffusing particles: We show that in suitable weak metrics, the law of fluctuations as predicted by the discretized Dean-Kawasaki SPDE approximates the law of fluctuations of the original particle system, up to an error that is of arbitrarily high order in the inverse particle number and a discretization error. In particular, the Dean-Kawasaki equation provides a means for efficient and accurate simulations of density fluctuations in weakly interacting particle systems.