We investigate two thermo-spintronic responses in proximity induced superconductors with spin-orbit coupled band - spin Nernst effect and thermal Edelstein effect stemming respectively from momentum-space and mixed superconducting Berry curvatures. We unveil that the Bloch band spin-orbit coupling and pairing are entangled in shaping the superconducting Berry curvatures rather than in an additive manner. The resulting thermo-spintronic responses are significantly enhanced at low temperatures by the emergence of Bogoliubov Fermi surface, and can be effectively controlled by tuning the latter. The effects render a probe of Bogoliubov Fermi surface and superconducting Berry curvatures supported by it. Our work reveals Berry curvature properties in topologically trivial proximity superconductors with spin-orbit coupling, and opens a new route to superconducting spintronics.