This paper develops a numerical approach to compute polar parity perturbations within fully relativistic models of black hole systems embedded in generic, spherically symmetric, anisotropic fluids. The authors apply this framework to study gravitational wave generation and propagation from extreme mass-ratio inspirals (EMRIs) in the presence of several astrophysically relevant dark matter (DM) models, including the Hernquist, Navarro-Frenk-White, and Einasto profiles. They also study dark matter spike profiles obtained from a fully relativistic calculation of the adiabatic growth of a black hole within the Hernquist profile and provide a closed-form analytic fit of these profiles. The analysis complements prior numerical work in the axial sector, yielding a fully numerical pipeline to study black hole environmental effects. The authors investigate the dependence of the fluxes on the DM halo mass and compactness, finding that polar fluxes are not adequately described by simple gravitational-redshift effects, unlike the axial case. This finding opens up exciting avenues for studying black hole environments with gravitational waves.This paper develops a numerical approach to compute polar parity perturbations within fully relativistic models of black hole systems embedded in generic, spherically symmetric, anisotropic fluids. The authors apply this framework to study gravitational wave generation and propagation from extreme mass-ratio inspirals (EMRIs) in the presence of several astrophysically relevant dark matter (DM) models, including the Hernquist, Navarro-Frenk-White, and Einasto profiles. They also study dark matter spike profiles obtained from a fully relativistic calculation of the adiabatic growth of a black hole within the Hernquist profile and provide a closed-form analytic fit of these profiles. The analysis complements prior numerical work in the axial sector, yielding a fully numerical pipeline to study black hole environmental effects. The authors investigate the dependence of the fluxes on the DM halo mass and compactness, finding that polar fluxes are not adequately described by simple gravitational-redshift effects, unlike the axial case. This finding opens up exciting avenues for studying black hole environments with gravitational waves.