Given $\mu, \nu \in \mathcal{P}(\mathbb{R})$ the space of positive measures on $\mathbb{R}$ we want to show that
$$ d(\mu, \nu) = \sup_{|f|\leq 1\\ f \ 1-\text{Lipschitz}}\left|\int f d\mu - \int fd\nu \right|$$
is a metric on $\mathcal{P}(\mathbb{R})$. It doesn't seem to me like the functions' properties are needed anywhere. Am I missing something?
To show that d is a metric you will have to show that $\int f d\mu=\int f d\nu$ for every 1-Lipschtz function f implies $\mu =\nu$. (Since both sides are linear in f the value of the Lipschitz constant doesn't matter). A very detailed argument would be lengthy so I will give a sketch. Let C be a compact set and $D=\{x :d(x,C) \geq 1/n\}$. Let $f_n(x)=\frac {d(x,D)} {d(x,C)+d(x,D)}$. Verify that this is indeed a Lipschtiz function. (This is a bit lengthy but some simple algebraic manipulation will give this). Show that $f_n \to I_C$ pointwise and apply DCT to see that $\mu (C)=\nu (C)$. Since C is arbitrary we get $\mu= \nu$