We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE$_{κ'}$ for $κ'$ in $(4,8)$ that is drawn on an independent $γ$-LQG surface for $γ^2=16/κ'$. The results are similar in flavor to the ones from our paper dealing with CLE$_κ$ for $κ$ in $(8/3,4)$, where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the CLE$_{κ'}$ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: Our previous paper "CLE percolations" described the law of interfaces obtained when coloring the loops of a CLE$_{κ'}$ independently into two colors with respective probabilities $p$ and $1-p$. This description was complete up to one missing parameter $ρ$. The results of the present paper about CLE on LQG allow us to determine its value in terms of $p$ and $κ'$. It shows in particular that CLE$_{κ'}$ and CLE$_{16/κ'}$ are related via a continuum analog of the Edwards-Sokal coupling between FK$_q$ percolation and the $q$-state Potts model (which makes sense even for non-integer $q$ between $1$ and $4$) if and only if $q=4\cos^2(4π/κ')$. This provides further evidence for the long-standing belief that CLE$_{κ'}$ and CLE$_{16/κ'}$ represent the scaling limits of FK$_q$ percolation and the $q$-Potts model when $q$ and $κ'$ are related in this way. Another consequence of the formula for $ρ(p,κ')$ is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.