In the present study, a new hybrid approach is proposed to modeling coupled vibrational and chemical kinetics in carbon dioxide (CO2) and products of its decomposition. The study develops and completes our previous work carried out for a single-component CO2 gas. The model is based on self-consistent implementation of state-to-state chemical and energy production rates into the equations of multi-temperature CO2 kinetics. It distinguishes vibrational temperatures of all CO2 modes and diatomic species and thus takes into account multiple relaxation mechanisms including intra-mode, inter-mode, and inter-molecular energy transitions as well as state-specific dissociation and exchange reactions. Other advantages of the proposed full multi-temperature approach are the possibility of capturing strong non-equilibrium effects in a flow, straightforward implementation of the chemical-vibrational coupling terms, easy update for new models of state-specific reaction rates. Comparisons with the results obtained in the frame of a detailed but numerically demanding state-to-state approach for the problem of spatially homogeneous relaxation showed good accuracy of the new model under the wide range of initial conditions; at the same time, traditional multi-temperature approaches failed to provide accurate predictions of non-equilibrium flow parameters under arbitrary deviations from equilibrium. Effects of chemical reaction models and selective mode excitation are assessed. The numerical efficiency of the developed model is found acceptable compared to that of the state-to-state approach.