Stars passing near the Sun (known as “Nemesis” stars) may play a significant role in the evolution of our solar system. Previous work has already searched for and characterized these stellar encounters using data from the Gaia astrometric survey and a time-independent model of the Galactic potential. However, other work has shown that including a rotating triaxial bar in the potential model, thus creating a time-dependent potential, significantly affects the orbits of stars in the solar neighborhood. We use three variations (a static bulge, a short-fast bar, and a long-slow bar) on two different best-fit Milky Way potential models (fit to different rotation-curve data), to examine the variation in the predicted closest-approach distance of Gaia DR2 Nemesis star candidates identified using an axisymmetric static model. Using short-term orbit integrations (±15 Myr) and Monte Carlo sampling of measurement uncertainties, we find that orbits of the Sun and Nemesis stars change more with different static models than they do with the introduction of rotating bars. Interestingly, we find that the Nemesis stars are significantly more sensitive to potential differences than a random sample of solar neighborhood stars. Using long-term orbit integrations (±10 Gyr), we find that this sensitivity may be connected to the presence of resonances spanned by the range of forecasted orbital frequencies for each star, which change drastically between models. This variation further indicates that properly modeled measurement uncertainties are crucial in characterizing stellar orbits.