Let $\mathcal X=(\mathcal X,d)$ be a Polish space equipped with the Borel sigma-algebra. Let $p\ge 1$ and $P_1,P_2$ be probability distributions on $\mathcal X$ such that $\max_{k=1,2}\int d(x,x_0)^pdP_k(x) < +\infty$ for some (and therefore for every!) $x_0 \in \mathcal X$. A fancy way to say this is that, $P_1$ and $P_2$ are in the $p$-Wasserstein space over $\mathcal X$, usually denoted $\mathcal P_p(\mathcal X)$.
Now, for every $\varepsilon \ge 0$ and coupling $(X,X')$ of $P_1$ and $P_2$, one can use Markov's inequality to get the bound
$$ \mathbb P(d(X,X') > \varepsilon) \le \mathbb P(d(X,X')^p > \varepsilon^p) \le \frac{\mathbb E[d(X,X')^p]}{\varepsilon^p}. $$
Optimizing over the coupling $(X,X')$, we get
$$\inf_{X,X'}\mathbb P(d(X,X') > \varepsilon) \le \left(\frac{W_p(P_1,P_2)}{\varepsilon}\right)^p. $$
Now, define $TV_\varepsilon(P_1,P_2) :=\inf_{X,X'}\mathbb P(d(X,X') > \varepsilon)$ and note that $TV_\varepsilon(P_1,P_2) \le TV_0(P_1,P_2) = \inf_{X,X'}\mathbb P(X \ne X')$ corresponds to the usual total-variation metric. Thus $TV_\varepsilon$ extends the usual total-variation metric. BTW, $TV_\varepsilon$ is a Wasserstein distance induced by the cost $c_\varepsilon(x,x') = \begin{cases}1,&\mbox{ if }d(x,x') > \varepsilon,\\0,&\mbox{ else.}\end{cases}$
We have proved that
[Weak bound] For every $\varepsilon > 0$ and $P_1,P_2 \in \mathcal P_p(\mathcal X)$, we have $$ TV_\varepsilon(P_1,P_2) \le \left(\frac{W_p(P_1,P_2)}{\varepsilon}\right)^p, \tag{1} $$
where $W_p(P_1,P_2)$ is the $p$-Wasserstein distance between $P_1$ and $P_2$. Of course, the above inequality is only meaningful for $\varepsilon \ge W_p(P_1,P_2)$. In particular, it breaks down completely for $\varepsilon > 0$. This issue comes at no surprise (at least to me!), as we have only imposed any important constraints on the distributions $P_1$, $P_2$.
Question (stronger bounds). How can the bound (1) be improved under further restrictions (for example, sub-Gaussianness, etc.) on the distributions $P_1$ and $P_1$ ?
For example, the holy grail for me would be to exhibit a function $F:[0, 1] \rightarrow [0, 1]$, such that $TV_\varepsilon(P_1,P_2) \le F(TV(P_1,P_2))$, uniformly on $P_1$ and $P_2$. Of course, I'm open to other kinds of bounds.
Notes
For simplicity, I've ignored issues related to the possible non-measurability of the set $\{(x,x') \in \mathcal X^2 \mid d(x,x') > \varepsilon\}$.
