Fix a prime $p$ and a constant $\frac{1}{p}<α\leq 1$. Consider the random directed Erdős--Rényi bipartite graph $\vec G(n,\lceilαn\rceil ,v)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceilαn\rceil$, and edge probability $0<v<1$. Bhargava, DePascale and Koenig conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $\vec G(n,\lceilαn\rceil,v)$ as $n\to\infty$. We prove this conjecture. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group onto $H$, for each finite abelian $p$-group $H$. However, in the case of $p$-Sylow subgroups of sandpile groups of random directed bipartite graphs, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this by restricting to a high-probability subset of graphs on which the surjective moments are well-behaved, and discarding a rare exceptional set of graphs whose contribution to the distribution vanishes but whose contribution to the surjective moments often diverges. Computing the conditional surjective moments on the good set and applying Wood's universality theorem yields the desired convergence in distribution.