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Strong anharmonicity in perovskites and other structurally complex semiconductors and oxides leads to temperature dependent phonon frequencies and phase transitions, both of which pose a challenge to computing electron-phonon scattering from first principles. Here, we show an approach to calculate temperature-dependent lattice vibrations and electron-phonon coupling in such strongly anharmonic crystals. Our method combines density functional perturbation theory (DFPT), the temperature dependent effective potential (TDEP) method, and a new approach to correctly include in the lattice dynamics the long-range dipole-dipole interactions. We apply this method to the canonical perovskite SrTiO₃, for which we compute the phonon dispersions, electron-phonon coupling (including the soft modes) and electron mobility for a range of temperatures spanning the tetragonal-to-cubic phase transition. Our work enables ab initio calculations of electron-phonon scattering and charge transport in perovskites and other anharmonic crystals with multiple phase transitions.