Radio detection of cosmic-ray (CR) induced extensive air showers with digital antenna arrays is a matured technique by now. At the Pierre Auger Observatory, the Auger Engineering Radio Array (AERA) has been measuring air-shower signals in conjunction with the particle detectors of the surface detector (SD) for over ten years. For an absolute determination of the CR energy with the Auger baseline detectors, the shower size estimator from the SD is calibrated with the energy scale of the fluorescence detector (FD). However, AERA has an independent access to the energy scale through the reconstructed radio signals. The hybrid detectors at the Pierre Auger Observatory offer the unique opportunity to compare the two independent energy scales. In this contribution, we present our envisaged methodology for cross-checking the agreement between the energy scales of the FD and AERA using hybrid SD-AERA shower data and simulations. We show individual steps of our radio signal reconstruction and highlight the key ingredients for calibrated energy measurements.