The tin-vacancy (SnV) center in diamond is a promising spin–photon interface for quantum networks, combining favorable optical properties with spin coherence above . Unfolding the full potential requires cavity enhancement to increase photon–emitter coupling efficiency. Here, we demonstrate cavity-enhanced light–matter coupling of SnV centers in a fully tunable Fabry–P'erot microcavity operating at temperatures down to with in-situ magnetic field control. We access the diamond-like regime of hybrid cavity modes through integration of low-roughness diamond membranes, where the field is concentrated inside the diamond and Purcell enhancement is maximized. Diamond-like modes deliver a more than two-fold increase in the effective Purcell factor over air-like modes, reaching $C_0=4.1(1)$ compared to $C_0=1.85(5)$ in the air-like case, while simultaneously relaxing mechanical stability requirements. Resonant probing reveals coherent cavity–emitter coupling with 96% extinction contrast and a coherent cooperativity of $C=4.0(13)$. By applying a magnetic field, we further achieve spin-resolved cavity extinction, observing spin-selective optical transitions with a projected contrast of ${𝒞}_{spin}=0.91$. These results establish SnV centers in diamond coupled to open Fabry–P'erot microcavities as a promising platform for efficient spin–photon interfaces.