Since you ask, there has been much research on the analogous question of elements of finite order in the [Nottingham group][1], which is the group of power series under composition:
$$ N(\mathbb K) = \left\{ x + \sum_{n\ge2} c_nx : c_n\in \mathbb K\right\}. $$
especially for $\mathbb K$ a finite field. You're example $-\frac{2x}{\sqrt{x^2-4}}$ is almost of this form, it's expansion starts $ix+\text{h.o.t.}$ See [this question][2], which is related to what you're asking about, and my answer to that question gives some references that discuss elements of finite order in the Nottingham group.


  [1]: https://en.wikipedia.org/wiki/Nottingham_group
  [2]: https://mathoverflow.net/q/425844