In a recent breakthrough result, Balliu et al. [FOCS'19] proved a deterministic $Ω(\min(Δ,\log n /\log \log n))$-round and a randomized $Ω(\min(Δ,\log \log n/\log \log \log n))$-round lower bound for the complexity of the bipartite maximal matching problem on $n$-node graphs in the LOCAL model of distributed computing. Both lower bounds are asymptotically tight as a function of the maximum degree $Δ$. We provide truly tight bounds in $Δ$ for the complexity of bipartite maximal matching and many natural variants, up to and including the additive constant. As a by-product, our results yield a considerably simplified version of the proof by Balliu et al. We show that our results can be obtained via bounded automatic round elimination, a version of the recent automatic round elimination technique by Brandt [PODC'19] that is particularly suited for automatization from a practical perspective. In this context, our work can be seen as another step towards the automatization of lower bounds in the LOCAL model.