As a note on a similar technique you are applying Mike but trying to keep the vibe much more fractional calculus'y (since it's what I am familiar with) We will try the following function:
We want 
fairly small
 = \sum_{n=0}^\infty \frac{w^n}{n!(^n e)})
We know that| < C e^{\kappa |w|})
for 
because } < C_\kappa \kappa^n)
So that > 0)
.
 = \frac{1}{\Gamma(z)}\int_0^\infty e^{-\lambda x}\beta(-x)x^{z-1} \,dx)
This function should be smaller then tetration at natural values.
 = \sum_{j=0}^n \frac{n!(-\lambda)^{n-j}}{j!(n-j)!(^j e)})
We would get the entire expression for)
by Lemma 3 of my paper:
 = \frac{1}{\Gamma(-z)}(\sum_{n=0}^\infty F(n)\frac{(-1)^n}{n!(n-z)} + \int_1^\infty e^{-\lambda x} \beta(-x)x^{z-1}\,dx))
Now F(z) will be susceptible to alot of the techniques I have in my belt involving fractional calculus. This Idea just popped into my head but I'm thinking working with a function like this will pull down the imaginary behaviour and pull down the real behaviour.
We also note that
 }\approx F(z+1) )
. Which again will be more obvious if you look at the paper, but it basically follows because:
 \approx (^n e))
We know that
So that
This function should be smaller then tetration at natural values.
We would get the entire expression for
Now F(z) will be susceptible to alot of the techniques I have in my belt involving fractional calculus. This Idea just popped into my head but I'm thinking working with a function like this will pull down the imaginary behaviour and pull down the real behaviour.
We also note that