In this paper we derive quantitative estimates in the context of stochastic homogenization for integral functionals defined on finite partitions, where the random surface integrand is assumed to be stationary. Requiring the integrand to satisfy in addition a multiscale functional inequality, we control quantitatively the fluctuations of the asymptotic cell formulas defining the homogenized surface integrand. As a byproduct we obtain a simplified cell formula where we replace cubes by almost flat hyperrectangles.