The book Profinite Groups of Ribes-Zalesskii shows the existence of free constructions using a similar ideia in all of them (free pro-C groups, free pro-C products, etc). In these proofs the author using the following fact:
"it is enough to check the universal for C-groups because a pro-C group is the inverse limit of C-groups".
I'm not sure if I'm able to verify this fact. For example, consider $(F,f)$ free pro-C group over $X$ and suppose that the universal property holds for C-groups. Let $G$ be any pro-C groups and decompose $$G = \varprojlim G_i$$ where $G_i$ are C-groups. Let $\pi_i: G \to G_i$ epimorphisms (we can assume a surjective inverse system). If $g: X \to G$, then $\pi_i g: X \to G_i$ are in the hypothesis, then there is a continuous map $\varphi_i: F \to G_i$ such that $\varphi_i f = \pi_i g$. Take $\varphi = (\varphi_i)$, then $$\varphi f = (\varphi_i f) = (\pi_i g) = g?$$ I'm not sure this is true.