Small-scale instabilities, which are local compared to the systems they exist in, play an instrumental role in understanding the mechanisms that lead to complex and often three-dimensional flow features. Vortices, for example, are ubiquitous in a turbulent flow and understanding their instabilities helps in a dynamical understanding of turbulence. A local stability approach, which calculates the inviscid evolution of relatively short-wavelength perturbations on a given base flow, has been instrumental in understanding various instabilities in vortical flows. In this study, we explore the effects of Schmidt number (Sc), which is the ratio between momentum and density diffusion coefficients, on the small-scale instabilities in an elliptical vortex with a stable stratification along its vortical axis. While the momentum and density diffusion coefficients are individually assumed to be small, their ratio Sc is allowed to be of arbitrary magnitude. The inviscid elliptical instability gets greatly modified due to the presence of a stable stratification. For Sc = 1, diffusion is shown to serve only as a suppressant of existing diffusion-free instabilities. We discover, however, that due to the presence of a stable stratification and a non-unity Sc, the vortex can be unstable in regimes which are stable based on a diffusion-free analysis. We characterize these non-unity Sc instabilities in detail, study how their growth rates depend on various base flow and perturbation parameters, and explore their potential connections with the diffusion-free instabilities