I denote a partition of an integer $n$ by $\vec i = (i_1, i_2, \ldots)$ (with $i_1, i_2, \ldots \in \mathbb N$) and define it by $$ \sum_{p\geq1} p i_p = n. $$ I set $$ |\vec i| = \sum_{p\geq1} i_p. $$ In my calculation, I've encountered the sum $$ \sum \binom{\ell}{\vec j, \ell - |\vec j|} \binom{k}{\vec i, k - |\vec i|} $$ where the sum is over integer $k\leq\ell\leq n$ and partitions of $n-\ell$ and $\ell-k$ named $\vec i, \vec j$ that also satisfies $\vec i + \vec j = (i_1 + j_1, i_2 + j_2, \ldots) = \vec r$ which is a fixed partition of $n-k = (n-\ell) + (\ell-k)$. I have used the notation of the multinomial coefficient $$ \binom{n}{k_1, \ldots, k_p} = \frac{n!}{k_1! \cdots k_p!}, \quad \text{when } k_1 + \ldots + k_p = n. $$ Note that some of the terms of the sum are zero because if one of the $k_i < 0$, then the multinomial coefficient is defined to be $0$.
There might exist a solution similar to the one proposed in this answer (as the $G$ term defined in the question matches the denominator of the multinomial coefficient), but I'm not familiar to generating functions over partitions enough to know if that's true or not. There is also a similarity with this answer.
