Radiation is an example of one of the techniques used for pasteurization and sterilization in various packaging systems. There is a high demand for the evaluation of the possible degradation of new composites, especially based on natural raw materials. The results of experimental research that evaluated the impact of radiation technology on biodegradable and compostable packaging materials up to 40 kGy have been presented. Two commercially available flexible composite films based on aliphatic–aromatic copolyesters (AA) were selected for the study, including one film with chitosan and starch (AA-CH-S) and the other with thermoplastic starch (AA-S). The materials were subjected to the influence of ionizing radiation from 10 to 40 kGy and then tests were carried out to check their usability as packaging material for the food industry. The results showed that the mechanical properties of AA-S films improved due to the radiation-induced cross-linking processes, while in the case of AA-CH-S films, a considerable decrease in the elongation at break was observed. The results also showed a decrease in the WVTR in the case of AA-S and no changes in barrier properties in the case of AA-CH-S. Both materials revealed no changes in the odor analyzed by sensory analysis. In the case of the AA-S films, the higher the radiation dose, the faster the biodegradation rate. In the case of the AA-CH-S film, the radiation did not affect biodegradation. The performed research enables the evaluation of the materials intended for direct contact with food. AA-CH-S was associated with unsatisfactory parameters (exceeding the overall migration limit and revealing color change during storage) while AA-S showed compliance at the level of tests carried out. The study showed that the AA-CH-S composite did not show a synergistic effect due to the presence of chitosan.