We study the weak KPZ universality problem by extending the KPZ universality
results for weakly asymmetric exclusion processes to non-simple variants under
deterministic initial data with constant hydrodynamical limit. We study a
random, microscopic field known as the \emph{density fluctuation field}
developed at microscopic stationarity by Goncalves-Jara \cite{GJ16},
Goncalves-Jara-Sethuraman \cite{GJS15} and Gubinelli-Perkowski \cite{GP}. The
particle dynamics of interest are of arbitrarily finite range, vastly improving
the result for range at most 3 obtained in \cite{DT}. The proof of this result
is the following novel two-step approach to study deterministic initial data.
First, we construct pseudo-stationary approximations, whose KPZ limits are
treated in aforementioned works, to deterministic initial data in the sense of
relative entropy. We then extend this to a global approximation in time, which
could be of independent interest for weakly asymmetric exclusion processes
beyond the universality problem. Second, this mechanism reduces the weak KPZ
universality problem into a compactness problem for linear SPDE, which is
directly treatable with classical stochastic analytic techniques.