Prokhorov metric vs. total variation norm

On metric spaces, the Prokhorov metric can always be bounded by the total variation metric. If $S$ is finite, you can bound the total variation metric by the Prokhorov metric via the Wasserstein metric.

More metrics and their relations (including the ones above) are nicely summariced in Gibbs A.L. and Su F.E., On Choosing and Bounding Probability Metrics, International Statistical Review 70 (2002), pp. 419-435.

Generally, knowing things about $\pi$ cannot tell you much about $\rho$; weak convergence is much easier to achieve than total variation convergence.

For example, if $S$ is any non-discrete space, $x$ is a limit point, and $x_n \to x$ is a sequence with $x_n \ne x$ for all $n$, then the point masses $\delta_{x_n}$ satisfy $\pi(\delta_{x_n}, \delta_x) \to 0$. (Indeed, $\pi(\delta_{x_n}, \delta_x) = d(x_n, x)$.) But we have $\rho(\delta_{x_n}, \delta_x) = 1$ for every $n$.