How large is the braid monodromy of low-genus Lefschetz fibrations?

Given a genus $g$ smooth Lefschetz fibration $\pi : M \to S^2$ with singular locus $\Delta \subseteq S^2$, we describe the subgroup $\operatorname{Br}(\pi)$ of the spherical braid group $\operatorname{Mod}(S^2,\Delta)$ consisting of braids admitting a lift to a fiber-preserving diffeomorphism of $M$. We develop general methods for showing that the index $[\operatorname{Mod}(S^2,\Delta) : \operatorname{Br}(\pi)]$ is infinite. As an application of our methods, we prove that $[\operatorname{Mod}(S^2,\Delta) : \operatorname{Br}(\pi)] = \infty$ when $g = 1$, when $\pi$ is expressible as a self-fiber sum when $g \geq 2$, or when $\pi$ is a holomorphic genus $g = 2$ Lefschetz fibration whose vanishing cycles are nonseparating. In the genus $g = 1$ case, we relate the subgroup $\operatorname{Br}(\pi)$ to the action of $\operatorname{Mod}(S^2,\Delta)$ on the $\operatorname{SL}_2$-character variety for $S^2 \setminus \Delta$ and provide an alternate proof of the first application via recent work of Lam--Landesman--Litt.