We present improved estimates of air-sea CO2 exchange over three latitude bands of the Southern Ocean using atmospheric CO2 measurements from global airborne campaigns and an atmospheric 4 - box inverse model based on a mass- indexed isentropic coordinate (M theta e). These flux estimates show two features not clearly resolved in previous estimates based on inverting surface CO2 measurements: a weak winter - time outgassing in the polar region and a sharp phase transition of the seasonal flux cycles between polar/subpolar and subtropical regions. The estimates suggest much stronger summer - time uptake in the polar/subpolar regions than estimates derived through neural- network interpolation of pCO2 data obtained with profiling floats but somewhat weaker uptake than a recent study by Long et al. [Science 374, 1275-1280 (2021)], who used the same airborne data and multiple atmospheric transport models (ATMs) to constrain surface fluxes. Our study also uses moist static energy (MSE) budgets from reanalyses to show that most ATMs tend to have excessive diabatic mixing (transport across moist isentrope, 0e, or M theta e surfaces) at high southern latitudes in the austral summer, which leads to biases in estimates of air-sea CO2 exchange. Furthermore, we show that the MSE-based constraint is consistent with an independent constraint on atmospheric mixing based on combining airborne and surface CO2 observations.