= Solution
Let $\widehat C|\Omega\rangle=|\Omega\rangle$. Since charge conjugation sends both $A_\mu$ and either Noether current to its negative and commutes with time ordering,
$$
\langle\Omega|T\mathcal O_1\cdots\mathcal O_n|\Omega\rangle
=(-1)^n\langle\Omega|T\mathcal O_1\cdots\mathcal O_n|\Omega\rangle.
$$
For odd $n$ the correlator equals its negative and hence vanishes. This operator argument is exact and does not use a perturbation expansion. It is <Furry's theorem>: scattering amplitudes with an odd number of external photons and no charged external particles vanish to every order, whereas even-photon scattering is allowed.
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