= Solution
First fix the source at the spatial origin. <Fourier transform> replaces $\partial_t$ by $-i\omega$ and $\nabla$ by $i\mathbf k$ in the free term. Multiplication by a static background becomes a <momentum> convolution; the <derivative> in $A\cdot\partial G$ acts on the <Green function>, so it carries the internal <momentum> $\mathbf k'$. Consequently
$$
\boxed{(k^2+m^2-\omega^2)G(\mathbf k;\omega)
+2e\int\frac{d^3k'}{(2\pi)^3}
[\omega A^0(\mathbf k-\mathbf k')-\mathbf A(\mathbf k-\mathbf k')\cdot\mathbf k']
G(\mathbf k';\omega)=1.}
$$
The transformed point source supplies the right-hand side one. The signs follow from $A^\mu\partial_\mu=A^0\partial_t+\mathbf A\cdot\nabla$ in these conventions.
\b[Source-position qualification.] For a nonuniform $A(\mathbf r)$, the general <Green function> is not <translation> invariant. In the printed basis $e^{i\mathbf k\cdot(\mathbf r-\mathbf r')}$, its coefficient must also depend on $\mathbf r'$, and the convolution contains an extra factor $e^{i(\mathbf k-\mathbf k')\cdot\mathbf r'}$. The displayed source equation is valid for $\mathbf r'=0$, or with the background itself expressed in coordinates centred on that source. Equivalently, a general spatial Fourier <Green function> has two <momentum> arguments and a momentum-delta-function source. Staticity alone permits the time-difference transform, not a general position-difference-only transform.
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