The McLerran-Venugopalan (MV) model is a Gaussian effective theory
of color charge fluctuations at small-$x$ in the limit of large valence
charge density, i.e. a large nucleus made of uncorrelated
color charges. In this work, we explore the effects of the first non-trivial
(even C-parity) non-Gaussian correction on the color charge density to
the MV model in SU(2) and SU(3) color groups in the non-perturbative regime.
We also compare our results to existing perturbative ones on
a lattice setup, where multi-point correlators of color charges can be
computed for fixed configurations. We investigate three different
choices for the renormalization of the couplings figuring in the
non-Gaussian small-$x$ action and find that one of them allows to
control the deviations from the MV model as one approaches the
continuum while the other two lead to a scenario where the small-$x$
action evolves towards a critical theory dominated by strong
non-Gaussian fluctuations regardless of the system size.