Fourier Transform of modified Bessel function of second kind

Question

Solving a Dirichlet problem with the method of separation of variables for the PDE $$(\partial_{xx}+\partial_{yy}+((1-2s)/y)\partial_{y})u(x,y)=0$$ in $\mathbb{R}\times(1,+\infty)$ with data on $\mathbb{R}\times\{1\}$, I came to the following problem in trying to find a representation for a solution in terms of a convolution with an appropriate kernel. Given $s\in(0,1)$ and $y>1$, calculate the Fourier transform of $$\xi\mapsto K_s(y|\xi|)/K_s(|\xi|),$$ where $K_s$ is the modified Bessel function of the second kind. I tried to reduce this problem in a convolution problem calculating separately the transform of $\xi\mapsto K_s(|\xi|)$ and $\xi\mapsto 1/K_s(|\xi|)$, using integral representation for $K_s$. This give some hope for the first one, but then I got loss in the calculations and for the second one have no idea how to procede... Losing the remaining hope in solving the problem with my own strength, I decided to ask help. Can anyone give a closed formula and show how to calculate it? Thanks in advance.