Let us say have a sequence of $n$ 2-$D$ random variables $X_i=(\varepsilon_i/\sqrt{n},i\varepsilon_{i}\sqrt{6}/n^{3/2})$, where $\varepsilon_{i}$ are independent random variables such that $\mathbb{P}(\varepsilon_i =\pm 1)=1/2$. Denote by $S_n$ their sum and take a $2$-dimensional Gaussian random variable $Z$ with the same covariance matrix as $S_n$. Any standard Berry-Esseen theorem (say by Gotze or Bhachattarya or Bentkus) gives us that for any convex set $C$ we have \begin{equation} |\mathbb{P}(S_n\in C)-\mathbb{P}(Z\in C)|\leq c\gamma, \end{equation} where gamma is the sum of third absolute moments of $X_i$, which in this case behaves like $n^{-1/2}$.

Question: is there a standard way to pass from the distance between distribution functions to expectations of Lipschitz functions? That is, suppose $f$ is Lipschitz, can we still bound

\begin{equation} |\mathbb{E}f(S_n)-\mathbb{E}f(Z)| \end{equation} in therms of $\gamma$? If so, does the same bound of magnitude $\gamma$ still apply?