Since you ask, there has been much research on the analogous question of elements of finite order in the Nottingham group, 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, 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.

Since you ask, there has been much research on the analogous question of elements of finite order in the Nottingham group, 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. Your example $-\frac{2x}{\sqrt{x^2-4}}$ is almost of this form, it's expansion starts $ix+\text{h.o.t.}$ See this question, 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.