  We construct, for all $g\geq 2$ and $n\geq 0$, a spectral sequence of
rational $S_n$-representations which computes the $S_n$-equivariant reduced
rational cohomology of the tropical moduli spaces of curves $\Delta_{g,n}$ in
terms of compactly supported cohomology groups of configuration spaces of $n$
points on graphs of genus $g$. Using the canonical $S_n$-equivariant
isomorphisms $\widetilde{H}^{i-1}(\Delta_{g,n};\mathbb{Q}) \cong W_0
H^i_c(\mathcal{M}_{g,n};\mathbb{Q})$, we calculate the weight $0$, compactly
supported rational cohomology of the moduli spaces $\mathcal{M}_{g,n}$ in the
range $g=3$ and $n\leq 9$, with partial computations available for $n\leq 13$.
