Dark matter subhalos are predicted to perturb stellar streams; stream morphologies and dynamics can, therefore, constrain the mass distribution of subhalos. Using FIRE-2 simulations of Milky Way–mass galaxies, we demonstrate that the presence of an LMC analog significantly changes stream–subhalo encounter rates. The LMC analog brings in many subhalos, increasing encounter rates for streams near the massive satellite by 10%–40%. Additionally, the LMC analog displaces the host from its center of mass (inducing reflex motion), which causes a north–south asymmetry in the density and radial velocity distributions of subhalos. This asymmetry, combined with the presence of LMC-analog subhalos, causes encounter rates at the same distance to vary by 50%–70% across the sky, particularly in regions opposite the LMC analog. Furthermore, the LMC analog induces a density wake in the host's dark matter halo, further boosting the encounter rates near the LMC analog. We also explore how stream orbital properties affect encounter rates, finding up to a 50% increase for streams moving retrograde to the LMC analog’s orbit in the opposite quadrant. Finally, we report the encounter rates for Milky Way streams within the context of our simulations, both with and without the presence of an LMC analog. The dependence of encounter rates on stream location, orbit, and their position relative to the LMC has important implications for where to search for streams with spurs and gaps in the Milky Way.