  We compute the weight 2 cohomology of the Feynman transforms of the cyclic
(co)operads $\mathsf{BV}$ and $\mathsf{HyCom}$, and the top$-2$ weight
cohomology of the Feynman transforms of $D\mathsf{BV}$ and $\mathsf{Grav}$.
Using a result of Giansiracusa, we compute, in particular, the top$-2$ weight
cohomology of the handlebody group. We compare the result to the top$-2$ weight
cohomology of the moduli space of curves $\mathcal{M}_{g,n}$, recently computed
by Payne and the last-named author. We also provide another proof of a recent
result of Hainaut-Petersen identifying the top weight cohomology of the
handlebody group with the Kontsevich graph cohomology.
