We prove a generalised second-order Boltzmann-Gibbs principle for conservative interacting particle systems on a lattice whose stationary measures are not of product type and not invariant under particle jumps. The result, which requires neither a spectral gap bound nor an equivalence of ensembles, extends the classical framework to settings with correlated invariant measures and is based on quantitative bounds for the correlation decay. As an application, we show that the equilibrium density fluctuations of the Katz-Lebowitz-Spohn model with a given choice of parameters converge, under diffusive scaling, to the stationary energy solution of the stochastic Burgers equation.