= Solution
Write $k=|\mathbf k_i|$ for the background wavenumber and $\psi_i=e^{i\mathbf k_i\cdot\mathbf r}$. The <Helmholtz equation> gives
$$
(\Delta+k^2)\psi_s=-k^2(n^2-1)\psi.
$$
The <Born approximation for scalar wave scattering> replaces $\psi$ in this source term by $\psi_i$. For the question's contrast $V=-k(n-1)$, also linearize $n^2-1\simeq2(n-1)$. Since $(\Delta+k^2)e^{ikR}/R=-4\pi\delta$, the outgoing scattered field is
$$
\boxed{\psi_s(\mathbf r)=-\frac{k}{2\pi}\int_D V(\mathbf r')
\frac{e^{ik|\mathbf r-\mathbf r'|}}{|\mathbf r-\mathbf r'|}
e^{i\mathbf k_i\cdot\mathbf r'}\,d^3r'.}
$$
Without the small-contrast linearization, replace $V$ by $-k(n^2-1)/2$ in this formula.
Apply the <Weyl plane-wave representation>. For an observation plane wholly above or below $D$, put $\sigma=+1$ or $-1$, respectively, and choose $m=\sqrt{1-p^2-q^2}$ with nonnegative imaginary part. Define $\widetilde V(\boldsymbol\kappa)=\int_DV(\mathbf r')e^{-i\boldsymbol\kappa\cdot\mathbf r'}d^3r'$. Substitution gives
$$
\boxed{\psi_s(\mathbf r)=-\frac{ik^2}{4\pi^2}\int\frac{dp\,dq}{m}
e^{ik(px+qy+\sigma mz)}
\widetilde V(kp-k_{i,x},kq-k_{i,y},\sigma km-k_{i,z}).}
$$
For the lower observation plane the vertical phase has the opposite sign. This follows from the absolute separation $|z-z'|$ in the Weyl identity; its version with $z-z'$ alone applies only above the sources.
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