  We obtain new calculations of the top weight rational cohomology of the
moduli spaces $\mathcal{M}_{2,n}$, equivalently the rational homology of the
tropical moduli spaces $\Delta_{2,n}$, as a representation of $S_n$. These
calculations are achieved fully for all $n\leq 10$, and partially -- for
specific irreducible representations of $S_n$ -- for $n\le 22$. We also present
conjectures, verified up to $n=22$, for the multiplicities of the irreducible
representations $\mathrm{std}_n$ and $\mathrm{std}_n\otimes \mathrm{sgn}_n$.
  We achieve our calculations via a comparison with the homology of
compactified configuration spaces of graphs. These homology groups are equipped
with commuting actions of a symmetric group and the outer automorphism group of
a free group. In this paper, we construct an efficient free resolution for
these homology representations, from which we extract calculations on
irreducible representations one at a time, simplifying the calculation of these
homology representations.
