I don't know if it is indeed doable but I am trying to compute analytically the convolution :
$$ f(\vec{r}) = \int \mathrm{d}\vec{r}_1 \cos(4\theta_1) \frac{K_0(\lambda \lVert \vec{r}-\vec{r}_1\rVert)}{r_1^2}$$
at least on the $x$-axis but I really don't know how to proceed and Mathematica doesn't seem to be helpful starting from this. Do you guys have any idea ?