Following a successfully quantization scheme previously developed in Ref. [R. Casana, M. M. Ferreira, Jr., and F. E. P. dos Santos, Phys. Rev. D 90, 105025 (2014).] for a parity-even gauge sector of the SME, we have established the Gupta-Bleuler quantization of a parity-odd and CPT-even electrodynamics of the standard model extension (SME) without the need for a small photon mass regulator. Keeping the photons massless, we have adopted the gauge-fixing condition: G(Aμ)=(∂0+κ0j∂j)(A0+κ0kAk)+∂iAi. The four polarization vectors of the gauge field are exactly determined by solving an eigenvalue problem, exhibiting birefringent second-order contributions in the Lorentz-violating parameters. They allow us to express the Hamiltonian in terms of annihilation and creation operators whose positivity is guaranteed by imposing a weak Gupta-Bleuler constraint, defining the physical states. Consequently, we compute the field commutation relation that has been expressed in terms of Pauli-Jordan functions modified by Lorentz violation whose light-cone structures have allowed us to analyze the microcausality issue.