I am looking for a reference for the following kind of results.

Let $\Gamma$ be the space of Lipschitz curves $\text{Lip}([0,1]; \mathbb R^d)$ equipped with the sup norm. Let $B$ be a Borel subset of $\Gamma$ for which the following property holds: for every $\gamma \in B$ there exists a function $h_\gamma \colon [0,1] \to [0,T]$ (for some $T>0$ fixed) which is non-decreasing and such that $$\tag{1} \frac{dh_\gamma}{dt}(t) = 0 \Rightarrow \frac{d \gamma}{dt}(t)=0. $$ Let $s_{\gamma}$ be (any) inverse of $h_\gamma$ ($s_\gamma$ be have jumps and there is arbitrarily defined, taking any value in the interval $(s_\gamma^-,s_\gamma^+)$); in view of the assumption (1) it turns out the the map $$ \tilde{\gamma}(r) := \gamma(s_\gamma(r) ) $$ is well defined and continuous. Thus we define a "reparametrization" map $$ R:\Gamma \to C([0,1]; \mathbb R^d) $$ by $R(\gamma)= \tilde \gamma$. If we consider the sup norm also on $C([0,1]; \mathbb R^d)$ we can formulate the

Question. Which is the regualarity of $R$? Is it Borel?

I am pretty sure the result is true and well known but I cannot fined any reference nor I am able to prove it in a clean and reasonably quick way. Any ideas? Thanks in advance.