There is an example in "Kuczma, Marek; Choczewski, Bogdan; Ger, Roman: Iterative functional equations" that (I believe, unfortunately I only have incomplete access to it at the moment, so I am not certain) shows that this does not have to exist in general (Ex 11.4.1).
However the example is preceeded by: 'we wish to exhibit two phenomena a) diffeomorphism with no (smooth) interative roots' b) [...something on convex functions...] so I am quite sure this answers what you are asking for.
The example is this:
Fix $0 < s < 1$ and $0 < x_0 < x_1 < 1$ such that $x_1 < \sqrt{s} x_0 / s$. Take any convex f \in C_1 (R)$ such that $f(x)=sx$ for $x \le x_0$ and f(x)=\alpha x +\betafor $x\ge x_1$ where $\alpha = (1-\sqrt{s}x_0)/(1-x_1)$ and\beta=$f \in C_1 (R)$ such that (\sqrt{s}x_0$ f (x) = sx $ for - x_1)/(1-x_1)`$x \le x_0$ and $ f(x) = \alpha x + \beta$ for $x\ge x_1$ where $\alpha = (1-\sqrt{s}x_0)/(1-x_1)$ and $\beta= (\sqrt{s}x_0 - x_1)/(1-x_1)$. Suppose that a function $\varphi: R \to R$ satisfies $\varphi \circ \varphi =f$ and $\varphi$ is $C_1$ or convex. [...] This leads to a contradiction.
Of course the book contains an argument why there is a contradiction.
Since the intersection of the assumptions of this example and your question seem non-empty, this shows that the answer to your question is 'no', and that there even does not exist a $C_1$ solution.
Possibly there are easier accessible sources or simpler arguments in your situation, but I don't known this.