= Solution
Multiplying the <QED Feynman rules> in fermion-line order gives, up to the common convention for $i\mathcal M$,
$$
\boxed{
\mathcal M=-e^2\bar u(p')\left[
\not\!\epsilon_{\rm out}^{,*}
\frac{\not p+\not q+m}{s-m^2}
\not\!\epsilon_{\rm in}
+
\not\!\epsilon_{\rm in}
\frac{\not p-\not q'+m}{u-m^2}
\not\!\epsilon_{\rm out}^{,*}
\right]u(p).}
$$
The two terms are both required by the <Ward identity>; replacing either external polarization by its photon momentum makes their sum vanish after using momentum conservation and the external <Dirac equations>.
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