Cauchy integral also for b< e^(1/e)?

[Kouznetsov]

But how to compute it on the imaginary axis?

for b< e^(1/e), the algorithm of the evaluation on the imaginary axis through the integral Cauchi equation is not yet ready.
While, you may recover the function from the asymptotic behavior.
I believe, it is even better than Cauchi.

For b=sqrt(2), the expansion is
F(z)=2- u - v_2 u^2 -v_3 u^3...
where
u=u(z)=exp((z-t)Q)
Substitute the expansion into equation F(z+1)=exp_b(F(z)).
Calculate the Taylor expansion of F(z+1)-exp_b(F(z)) with respect to u,
Taking into account that u(z+1)=Qu(z).
Set to zero the coefficients in this expansion.
These equations determine Q, v_2, v_3, ...
Recover t from the condition F(0)=1.
Then, using F(z-1)=log_b(F(z)) ,
you can evaluate F everywhere, not only along the imaginary axis.

You may use the representation through the asymptotic for evaluation of few first derivatives. If you want a preise evaluation of many coefficients of the Taylor expansion, better, use the Cauchi; you may evaluate a hundred coefficients within a second.