= Solution
Choose the polar axis along $\mathbf r_1-\mathbf r_2$. Then
$$
\int e^{-ik\hat{\mathbf r}\cdot(\mathbf r_1-\mathbf r_2)}d\Omega
=2\pi\int_{-1}^1e^{-ikdu}du=4\pi\frac{\sin(kd)}{kd}.
$$
Using the <angular interference integral for two point scatterers>, integrating the two self terms and the <interference> terms gives
$$
\boxed{\sigma_{\rm tot}=\frac{4\pi}{k^2}\left[|c_1|^2+|c_2|^2+2\operatorname{Re}(c_1c_2^*)\frac{\sin(kd)}{kd}\right]}.
$$
For $kd\ll1$, the amplitudes add coherently and $\sigma\to4\pi|c_1+c_2|^2/k^2$; when $kd\gg1$ the angle-integrated <interference> is suppressed. These limits hold for fixed incoming direction; no incoming-orientation average has been silently imposed.
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