= Solution
The first scattered correction to the incident wave is
$$
\psi^{(1)}(\mathbf r)=-C\int d^3r'\,
\frac{e^{ik|\mathbf r-\mathbf r'|}}{|\mathbf r-\mathbf r'|}
V(\mathbf r')e^{i\mathbf k_i\cdot\mathbf r'}.
$$
Inserting it into the amplitude gives a second, positive contribution because the two minus signs multiply:
$$
\boxed{f^{(2)}=C^2\int d^3r\,d^3r'\,
e^{-i\mathbf k_f\cdot\mathbf r}V(\mathbf r)
\frac{e^{ikd}}dV(\mathbf r')e^{i\mathbf k_i\cdot\mathbf r'}}.
$$
Thus the amplitude through second Born order is $f^{(1)}+f^{(2)}+O(V^3)$. The intervening <outgoing Green function for the three-dimensional Helmholtz equation> describes propagation between the two interactions; replacing it by a real standing-wave kernel would discard the imaginary part needed for <unitarity>.
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