To detect ultra-high-energy neutrinos, experiments such as ARA and RNO-G target the radio emission these particles induce when cascading in the ice, using deep antennas in South Pole or in Greenland. One of the main backgrounds for such signals is the radio emission generated by cosmic-ray showers, either directly in the ice, or in the air and transmitted to the ice, which can both reach the deep antennas. The first detection of cosmic rays with deep antennas would thus validate this detection principle and allow us to calibrate the detectors. FAERIE, the Framework for the simulation of Air shower Emission of Radio for in-Ice Experiments, is a numerical tool that couples both CoREAS and GEANT4 Monte-Carlo codes to simulate the radio emission from cosmic-ray showers deep in the ice. Using this code, we will investigate cosmic-ray radio signatures and the possible implications on the design of a cosmic-ray veto.