Let's consider the rational functions whose numerator and denominator of the function term are coprime.
Which kinds of rational functions of one variable have an inverse relation that contains a branch that is a rational function?
Which kinds of polynomial functions of one variable have an inverse relation that contains a branch that is a rational function?
I assume the degree of the numerator and the degree of the denominator of the function term has to be less than or equal to $1$. Some calculations with algebraic equations with undetermined parameters as coefficients seem to show that. But I'm not sure.
If $R$ is a rational function and $S$ a branch of $R^{-1}$ in an open set $U \subset \Bbb C$ then $S(R(z)) = z$ in $U$. If $S$ is also a rational function then it follows that $S(R(z)) = z$ globally (as meromorphic functions).
It follows that $S$ is injective and therefore has degree one. Then $R$ has degree one as well.
So the only rational functions with a (local) rational branch of the inverse are rational functions of degree one (which are the Möbius transformations).