It is possible to derive Coulomb's potential from QED by considering the scattering process $e^-+e^-\rightarrow e^-+e^-$. The argument can be found in Peskin and Schroeder p.125, and it is reproduced in this stackexchange answer. The authors state that it is necessary to assume that the particles are distinguishable, meaning that only the $t$-channel diagram is included, and not the $u$-channel diagram. Why is this the case? Indistinguishability is of course not unique to QFT (i.e. it is a QM effect also) so I don't understand why it would be ignored in this limit. I expect that it has something to do with the fact that one electron is treated as fixed, but I don't quite follow how.
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