We study resonant heterodyne up-conversion in the RADES-BabyIAXO haloscope as a method to search for low-mass dark matter axions using microwave cavities. Starting from axion electrodynamics, we derive the axion-induced source term and the power extracted through a readout mode, explicitly accounting for the finite axion linewidth. This leads to effective quality factors that determine the pump–axion mixing, detection bandwidth, and detected signal power. We extend the BI-RME 3D full-wave formulation to heterodyne axion detection in a realistic two-port cavity, including pump leakage into the readout channel. Applying the formalism to the largest RADES-BabyIAXO cavity identifies the ${quasi\mathrm{\text{-}}TE}_{011}-{quasi\mathrm{\text{-}}TM}_{010}$ mode pair as a favorable configuration, enabling sensitivity to axion frequencies between 0.9 and 34.6 MHz. Analytical and full-wave predictions show excellent agreement at resonance, while the full-wave model provides a more accurate description off resonance and allows a precise characterization of the pump leakage. We also derive the optimal port couplings that maximize the scanning rate. Upper-limit sensitivity for cryogenic copper and superconducting niobium cavities indicate that, under thermal-noise-limited conditions and assuming sufficient pump-leakage rejection, the experiment could probe axion-photon couplings down to ${10}^{-15}{GeV}^{-1}$ at 90% confidence level, representing a significant improvement over previous heterodyne-based searches.