= Solution
Taking $q\to0$ in the <Ward-Takahashi identity> gives
$$
V_3^\mu(0,p,p)=e\,\frac{\partial G^{-1}(p)}{\partial p_\mu}
$$
up to the displayed Euclidean $i$ conventions. The ultraviolet divergence of the zero-momentum vertex is therefore exactly the derivative of the fermion self-energy divergence. In renormalization-constant notation this is
$$
\boxed{Z_1=Z_2},
$$
so the vertex and fermion <wave-function renormalization> are not independent. The remaining <charge renormalization> is controlled by photon wave-function renormalization; in a common convention $e_0=Z_3^{-1/2}e$.
The derivation used only an exact change of variables, invariance of the action, and invariance of the measure. It therefore holds nonperturbatively whenever the regulator and definition of the theory preserve the vector symmetry. A symmetry-breaking regulator requires symmetry-restoring <counterterms>; a genuine <quantum anomaly> would obstruct the identity, but vector QED has no such anomaly.
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