I think I might have an idea about what's going on here. This shows the result of graphing in the complex plane. The x-scale (real part) runs from 2 to 10 and the y-scale (imag part) runs from -8i to 8i. This function was obtained via the Cauchy integral (Kouznetsov's construction):

This, now, is the result of graphing over the same parameter range, obtained via the regular iteration in the "repelling" fashion (the limit formula converges dog slowly, by fortunately we don't need much precision to get a good graph). Both have value at 2:

There should be more detail in the white regions than is seen here (especially toward the right, the leftmost bits are correct), indeed, the behavior there is extremely complicated, but I can't get that, as the computer overflows when iterating the recurrence equation to draw the graph. So I set a "bail out" to suppress this, yet that has the effect of losing detail. But still, these graphs can be useful in examining the behavior of the functions involved.

As one can easily see, they can't even be translated to fit each other, not even in the asymptotic. Though they seem like they're asymptotic on the real line, on the complex plane, they're very different, which may be why the constructed "base change tetrational" is not analytic (Taylor series has 0 convergence radius). Smooth functions, analytic nowhere, are well known, and it is also well known that in Real analysis, a sequence of real-analytic functions can converge pointwise to a non-analytic function, just look at Fourier series, where the convergent may not even be continuous!

In addition, the limits at imaginary infinity are different. For the tetration, we have

while for the cheta function, we have

.

This, now, is the result of graphing over the same parameter range, obtained via the regular iteration in the "repelling" fashion (the limit formula converges dog slowly, by fortunately we don't need much precision to get a good graph). Both have value at 2:

There should be more detail in the white regions than is seen here (especially toward the right, the leftmost bits are correct), indeed, the behavior there is extremely complicated, but I can't get that, as the computer overflows when iterating the recurrence equation to draw the graph. So I set a "bail out" to suppress this, yet that has the effect of losing detail. But still, these graphs can be useful in examining the behavior of the functions involved.

As one can easily see, they can't even be translated to fit each other, not even in the asymptotic. Though they seem like they're asymptotic on the real line, on the complex plane, they're very different, which may be why the constructed "base change tetrational" is not analytic (Taylor series has 0 convergence radius). Smooth functions, analytic nowhere, are well known, and it is also well known that in Real analysis, a sequence of real-analytic functions can converge pointwise to a non-analytic function, just look at Fourier series, where the convergent may not even be continuous!

In addition, the limits at imaginary infinity are different. For the tetration, we have

while for the cheta function, we have

.