The impact of radiation reaction and Breit-Wheeler pair production on the acceleration of fully ionized carbon ions driven by an intense linearly polarized laser pulse has been investigated in the ultra-relativistic transparency regime. Against initial expectations, the radiation reaction and pair production at ultra-high laser intensities are found to enhance the energy gained by the ions. The electrons lose most of their transverse momentum, and the additionally produced pair plasma of Breit-Wheeler electrons and positrons co-streams in the forward direction as opposed to the existing electrons streaming at an angle above zero degree. We discuss how these observations could be explained by the changes in the phase velocity of the Buneman instability, which is known to aid ion acceleration in the breakout afterburner regime, by tapping the free energy in the relative electron and ion streams. We present evidence that these non-classical effects can further improve the highest carbon ion energies in this transparency regime.