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.
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