We consider inhomogeneous percolation on a hierarchical configuration model with a heavy-tailed degree distribution. This graph is the configuration model where all the half-edges are colored either black or white, and edges are formed by uniformly matching edges of the same color. When only the white half-edges are paired, we provide sufficient conditions for the size and total number of incident black half-edges of the connected components to converge in an $\ell^2$-sense. The limiting vector is described by an $\mathbb{R}^2$-valued thinned Lévy process. We also establish an $\ell^2$-limit for the number of vertices in connected components when a critical proportion of the black edges are included. A key part of our analysis is establishing a Feller-type property for the multiplicative coalescent with mass and weight recently studied in (Dhara et. al 2017, Dhara et. al 2020).