I am looking for some integral representation of the modified Bessel function of the second kind of order $0$, $K_0(z)$, where $z$ isn't on the real line. I need the only place that $z$ appears in this representation to be something like the form $\exp(-\alpha z^2)$ (for some $\alpha$ independent of $z$).
For context, I am trying to evaluate a ugly expression of the form $$\int_0^1 dx\sum_{\vec{k}\in\mathbb{Z}^2}\Re K_0\left(2\pi a \|\vec{k}\|\sqrt{m^2-x(1-x)s-i \varepsilon}\right)$$ with $a,m$ positive real numbers, $s>4m^2$ and $1\gg\varepsilon>0$. I have tried most of the obvious approaches such as Poisson summation, umbral methods and converting to $Y_0$, which all failed for a variety of reasons. I currently believe the best route forward would be to get the representation mentioned above, since I can evaluate the 2D sum of the exponential of the square of the magnitude of $\vec{k}$. I have searched in standard integral tables without much luck. That would leave me with two integrals that I hope to have good enough numerical properties, which other forms I have found does not.
Any other, perhaps more obscure, approaches would also be helpful, but isn't the main question.