Understanding the superfluid stiffness Dₛ is a central problem in flat-band superconductivity. It is often interpreted together with spectroscopic probes such as tunneling, since both ultimately reflect the same superconducting quasiparticles. Their relationship is nevertheless complicated in flat bands, where quantum-geometric contributions to Dₛ must be included. The situation in moiré graphene is particularly complex: tunneling experiments in some cases reveal an evolution between V- and U-shaped spectra together with a quite universal finite zero-bias conductance (ZBC). These two phenomena along with others have been argued to suggest the presence of a Bogoliubov Fermi surface (BFS), often generated by finite-momentum pair-density-wave (PDW) superconductivity. In this paper we calculate the superfluid stiffness in the presence of such a PDW. Virtual interband processes provide the familiar positive geometric contribution that gives phase rigidity to the flat-band condensate, allowing superconductivity to survive even in the presence of a BFS. At the same time, the multiband PDW pair structure enables the gapless BFS quasiparticles to respond to a phase gradient despite the negligible ordinary flat-band velocity. Their resulting counterflow reduces the stiffness even at zero temperature. As an experimentally accessible consequence, we predict a correlated evolution of the residual ZBC and the low-temperature stiffness: enhanced zero-bias spectral weight should accompany reduced phase rigidity. Observation of this correlation would support the presence of a BFS and indicate that the residual zero-energy states are intrinsic.