I have shown that $$\mathcal{F}_y\left\{\frac{e^{-|y|}}{y}\right\} = -i \sqrt{\dfrac{2}{\pi}} \tan ^{-1} (k)$$ where $k$ is the frequency variable.
I need to find, however, $$\mathcal{F}_y\left\{\frac{e^{-|y|x}}{y}\right\},$$ which I'm pretty sure is $-i \sqrt{\dfrac{2}{\pi}} \tan ^{-1} \left(\frac{k}{x}\right)$, but this is just intuition and I can't seem to find any properties of the Fourier transform to help me out.
If I'm pointed in the right direction I should be able to finish it myself.
You can use the standard identity
$$\mathcal{F}\{f(xy)\}=\frac{1}{|x|}F(k/x)\tag{1}$$
where $F(k)$ is the Fourier transform of $f(y)$. This of course only works in your case if $x>0$, but I guess this should be the case anyway, because otherwise the Fourier transform does not exist. Using (1) with $x>0$ will indeed give you the expected result.