https://math.stackexchange.com/questions/4829146/tetration-uniqueness-by-a-inf-sum-n-a-n2
My friend posted this question of mine on mathstackexchange.
More research is needed.
update
That sum we want to see minimized can be expressed by some integral of an integrand that contains the considered tetration function. The integral consideres the value of tetration at the unit circle and is thus in the spirit of cauchy contour integrals.
If this is helpful or not is another matter.
Maybe more later.