The coupling of chiral superfluids and superconductors to the background geometry leads to surprising geometric induction phenomena. We show that this coupling bears important consequences even in a nearly flat background, through its signature in thermal fluctuations. Starting from the Ginzburg-Landau free energy of a chiral superfluid minimally coupled to the background geometry, we show that the interaction strength between vortices gets renormalized by geometry fluctuations. In our setup, these arise from the shape fluctuations of the underlying two-dimensional substrate, and are controlled by its bending rigidity and tension. In the tensionless limit, the fluctuations lower the vortex interaction strength at large distances, which leads to a lowering of the BKT transition temperature. We study this effect in terms of the renormalization group flow of a dual sine-Gordon theory of the superfluid transition coupled to the substrate shape. It shows that, in turn, the chiral superfluid order can suppress the amplitude of shape fluctuations, resulting in an extended phase diagram which links the superfluid transition to the crumpling transition of the substrate. These reveal thermodynamic signatures of chirality in the superfluid transition.