On the existence of rational operators

[bo198214]

But, if tetration is not linear over domain [0, 1] rational operators depend on a logarithm base for their identity--and it no longer has universality.

Ya that was my first feel of your construction: "piecewise".
I mean its better than nothing, but usually one wants a smooth or better analytic function. I.e. that the function x {y} z is analytic in all 3 arguments.
Usually piecewise constructions aren't analytic at the gluing points (i.e. here at the integers).
Nonetheless its the first construction I heard of.
The requirements to such operators were discussed already on the board somewhere (perhaps later I can find these threads).