Is there some canonical way to approach integrals of type
$$ I(k,q) = \int {\rm d}^{3} s~ e^{i k \cdot s} f\left(|s|\right)g\left(|q-s|^2\right), $$
where $s$, $k$ and $q$ are momentum vectors, and $f$ and $g$ are simple, scalar, nice enough functions? This simplifies to an ordinary convolution in both $k\to 0$ or $r\to0$ limits and is then easy to handle.
Best I was able to get, using plane wave expansion, looks like $$ I(k,q) \propto \sum_{l} i^l(2l+1) \mathcal P_{l}\left(\hat k \cdot \hat r \right) \int \frac{s^2 ds}{2\pi^2}\frac{p^2 dp}{2\pi^2}~ j_l(p s) j_l(s k) j_l(p q) g(s) \tilde f(p), $$ where $\tilde f$ is the simple FT of $f$ and $P_{l}$ are Legendre polynomial . But the double integral here looks quite nasty.
The optimal goal would be to transform the problem into a sequence of 1D integrals. Is that possible?