Let $X$ be a compact metric space, and fix an arbitrary point $x_\ast \in X$. By the Kantorovich-Rubinstein duality theorem, the $1$-Wasserstein metric $W_1$ on the set of Borel probability measures on $X$ can be expressed as $$ W_1(\mu,\nu) = \max\left\{ \int f \, d\mu - \int f \, d\nu \, : \, f \in \mathcal{K}_0\right\}\text{,} $$ where $\mathcal{K}_0$ is the compact set (in the topology of uniform convergence) consisting of all $1$-Lipschitz functions $f \colon X \to \mathbb{R}$ with $f(x_\ast)=0$.
I now want to move from the purely topological setting to a "smooth" setting.
Suppose $X$ is the closure of a bounded convex open subset of $\mathbb{R}^N$, with the boundary being an $(N-1)$-dimensional $C^\infty$ embedded submanifold of $\mathbb{R}^N$. For each $r \in \mathbb{N}$, let $C^r(X,\mathbb{R})$ be the set of functions $f \colon X \to \mathbb{R}$ admitting a $C^r$ extension to an $\mathbb{R}^N$-neighbourhood of $X$.
For each $r \in \mathbb{N}$, does there exist a set $\mathcal{K}_r \subset C^r(X,\mathbb{R})$ that is compact in the $C^0$ topology on $X$ (i.e. topology of uniform convergence) such that $W_1$ can be expressed as
$$ W_1(\mu,\nu) = \max\left\{ \int f \, d\mu - \int f \, d\nu \, : \, f \in \mathcal{K}_r\right\} \text{?} $$