  Recent experimental advances in using strain engineering to significantly
alter the band structure of moderately correlated systems offer opportunities
and challenges to weak-coupling renormalization group (RG) analysis approaches
for predicting superconducting instabilities. On one hand, the RG approach can
provide theoretical guidance. On the other hand, it is now imperative to better
understand how the predictions of the RG approach depends on microscopic and
non-universal model details. Here we focus on the effect of band-selective
mass-renormalization often observed in angle resolved photoemission
spectroscopy. Focusing on a specific example of uniaxially strained
$\rm{Sr_2RuO_4}$ we carry out the weak-coupling RG analysis from two sets of
band structures as starting points: one is based on density functional theory
(DFT) calculations and the other is based on angle-resolved photoemission
spectroscopy (ARPES) measurements. Despite good agreement between the Fermi
surfaces of the the two band structures we find the two sets of band structures
to predict qualitatively different trends in the strain dependence of the
superconducting transition temperature $T_c$ as well as the dominant channel.
