Constant together

If $F(k, m,n)= \int_0^{\infty}\int_0^{\infty}\int_0^{\infty}\frac{kmn e^{\pi(x+y+z)} dx dy dz}{(e^{2\pi x}+1)(e^{2\pi y}+1)(e^{2\pi z}+1)(n^2+4x^2)(k^2+4y^2)(m^2+4z^2)} ,$ and $\lim_{k\to 0}\lim_{m \to 0} \lim_{n\to \infty} \left(1-\frac{d}{dm}F(k,m,n)\sum_{i=1}^{n}\sum_{j=1}^{i+1}\gamma^{i-j+2}\frac{\zeta(j+1)-1)}{\Gamma(i-j+1)}\right)^{qn}\cdot J=e^{e^{\gamma} \pi G}$ where $k,m,n$ are positive real numbers, $q$ and $J$ are positive integers, then find the value of $J+q$ .

Notation: $\gamma$ and $G$ denote Euler-Mascheroni constant and Catalan's constant .

This problem is the variant version of A general integral and also it is shared proposed problem in Romanian Mathematical Magazine to prove the closed form .