  Under in-plane uniaxial stress, the largest Fermi surface sheet of the
correlated metal Sr$_2$RuO$_4$ undergoes a Lifshitz transition from an
electron-like to an open geometry. We investigate the effects of this
transition on transport through measurement of the longitudinal resistivity
$\rho_{xx}$ and the Hall coefficient $R_\text{H}$. At temperatures where
scattering is dominated by electron-electron scattering, $R_\text{H}$ becomes
more negative across the Lifshitz transition, opposite to expectations from the
change in Fermi surface topology. We show that this change in $R_\text{H}$ is
explainable only if scattering changes throughout the Brillouin zone, not just
at the point in $k$-space where the Lifshitz transition occurs. In a model of
orbital-dependent scattering, the electron-electron scattering rate on sections
of Fermi surface with $xy$ orbital weight decreases dramatically. On the other
hand, at temperatures where defect scattering dominates $\rho_{xx}$ and
$R_\text{H}$ are essentially constant across the Lifshitz transition.
