2002cx-like supernovae are a sub-class of sub-luminous Type Ia supernovae (SNe). Their light curves and spectra are characterized by distinct features that indicate strong mixing of the explosion ejecta. Pure turbulent deflagrations have been shown to produce such mixed ejecta. Here, we present hydrodynamics, nucleosynthesis and radiative-transfer calculations for a 3D full-star deflagration of a Chandrasekhar-mass white dwarf. Our model is able to reproduce the characteristic observational features of SN 2005hk (a prototypical 2002cx-like supernova), not only in the optical, but also in the near-infrared. For that purpose we present, for the first time, five near-infrared spectra of SN 2005hk from -0.2 to 26.6 d with respect to B-band maximum. Since our model burns only small parts of the initial white dwarf, it fails to completely unbind the white dwarf and leaves behind a bound remnant of ~1.03Mconsisting mainly of unburned carbon and oxygen, but also enriched by some amount of intermediate-mass and iron-group elements from the explosion products that fall back on the remnant.We discuss possibilities for detecting this bound remnant and how it might influence the late-time observables of 2002cx-like SNe.