I found a proof for $M2=\sum _{n=1}^{\infty } \left(\frac{i}{\pi }\right)^{1-n}\left(\text G_{n+1,n+2}^{n+2,0}\left(^{1;\underbrace {x,…,x}_n}_{1-n,0; \underbrace {x,…,x}_n}\bigg|-\pi i\right)\right). $

It was so simple, it looked trivial, but it is indeed omnipotent!

In Mathematica, I entered

f[n_] := MeijerG[{{},Table[1, {n + 1}]}, {Prepend[Table[0, n + 1], -n + 1], {}}, -I \[Pi]];
Integrate[E^(I Pi x) (x^(1/x) - 1), {x, 1, Infinity I}]

and got exactly $$\int_1^{i \infty } e^{i \pi x} \left(x^{1/x}-1\right) \, dx.$$

Your analysis is still desired.

So, how do we code this? the facts are we have

$M2=\sum _{n=1}^{\infty } \left(\frac{i}{\pi }\right)^{1-n}\left(\text G_{n+1,n+2}^{n+2,0}\left(^{1;\underbrace {x,…,x}_n}_{1-n,0; \underbrace {x,…,x}_n}\bigg|-\pi i\right)\right).$

and this

found here