© 2022 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the "https://creativecommons.org/licenses/by/4.0/"Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP3.The double copy of the Coulomb solution in three dimensions is a nonvacuum solution that can be obtained through different matter couplings. It is the static black hole solution of Einstein-Maxwell theory or general relativity minimally coupled to a free scalar field (with one ghost sign in the action in both cases). We consider generalizations of these matter couplings by paying particular attention to the regularity of the static black solution on the gravity side and the corresponding single copy electric field in the gauge theory. We show that (i) Einstein-Born-Infeld theory yields a singular double copy, which admits stable orbits for certain choices of parameters, with a regular single copy electric field, and (ii) black hole solutions constructed by Bueno et al. [Phys. Rev. D 104, L021501 (2021).PRVDAQ2470-001010.1103/PhysRevD.104.L021501] by coupling to the scalar field exemplify mostly regular double copies with regular single copy electric fields and also admit stable orbits. Additionally, we use these solutions to investigate the connection between horizons on the gravity side and electric fields on the gauge theory side, which was previously observed in four dimensions.