Let $X_1,X_2\dots X_n$ be $n$ i.i.d $\mathbb{R}^d$-valued random variables such that $\mathbb{E}[X_i] = 0$ and $\text{Var}(X_i) = \frac{1}{n}I_{d\times d}$. From Central Limit Theorem (CLT) we know that the distribution of the sum $W = \sum_{i=1}^nX_i$ approaches $\mathcal{N}(0, I_{d\times d})$. I'm interested in understanding the approximation error of CLT. From Berry Esseen bounds we know that for any convex measurable set $A$ $$|\mathbb{P}(W\in A) - \mathbb{P}(Z\in A)| \leq c n d^{1/4} \mathbb{E}[\|X_i\|_2^3],$$ for some universal constant $c$, where $Z\sim\mathcal{N}(0, I_{d\times d})$. But I'm interested in a different kind of approximation error. Let $f:\mathbb{R}^d\to\mathbb{R}$ be a Lipschitz continuous function. I'd like to know how small the following quantity is $$\|\mathbb{E}[Wf(W)] - \mathbb{E}[Zf(Z)]\|_2.$$ Assuming $\mathbb{E}[\|X_i\|_2^3] = O(n^{-3/2})$, does this quantity go down with $n$?
P.S. I'm not too familiar with CLT and Stein's method. But from my limited understanding, it looks like this could be related to Stein's method. Any references on this would be greatly appreciated.