A recent study reported a strong apparent depression of Fe I, relative to Fe II, in the AGB stars of NGC 6752. This depression is much greater than that expected from the neglect of non-local thermodynamic equilibrium effects, in particular the dominant effect of overionisation. Here we attempt to reproduce the apparent Fe discrepancy, and investigate differences in reported sodium abundances. We compare in detail the methods and results of the recent study with those of an earlier study of NGC 6752 AGB stars. Iron and sodium abundances are derived using Fe I, Fe II, and Na I lines. Various uncertainties are explored. We reproduce the large Fe I depression found by the recent study, using different observational data and computational tools. Further investigation shows that the degree of the apparent Fe I depression is strongly dependent on the adopted stellar effective temperature. To minimise uncertainties in Fe I we derive temperatures for each star individually using the infrared flux method (IRFM). We find that the $T_{rm{eff}}$ scales used by both the previous studies are cooler, by up to 100 K; such underestimated temperatures amplify the apparent Fe I depression. Our IRFM temperatures result in negligible apparent depression, consistent with theory. We also re-derived sodium abundances and, remarkably, found them to be unaffected by the new temperature scale. [Na/H] in the AGB stars is consistent between all studies. Since Fe is constant, it follows that [Na/Fe] is also consistent between studies, apart from any systematic offsets in Fe. We recommend the use of $(V-K)$ relations for AGB stars. We plan to investigate the effect of the improved temperature scale on other elements, and re-evaluate the subpopulation distributions on the AGB, in the next paper of this series. [abridged]
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