The results of the author and Gess [27] develop a robust well-posedness theory for a broad class of conservative stochastic PDEs, with both probabilistically stationary and non-stationary Stratonovich noise, and with irregular noise coefficients like the square root. However, one case left untreated by [27] is the case of SPDEs that combine conservative, non-stationary Stratonovich noise with square root-like nonlinearities. Such equations arise naturally in the fluctuating hydrodynamics of inhomogenous systems, and a new analysis is required to handle certain discontinuous coefficients appearing in their Itô formulations. We treat the discontinuities by showing that the equation exhibits a novel regularization of the logarithm of the solution, and establish the well-posedness by building on the concept of a stochastic kinetic solution introduced in [27].