Suppose that I have $n$ dependent random variables $X_1,\ldots,X_n$ with $\mathbb{E}[X_i]=0, \mathbb{E}[X_i^2]=1$, where we have the following bounds on the Kolmogorov distance from a normal distribution: $$|P(X_i\leq x)-\Phi(x)|\leq \rho(x).$$ Here $\rho(x)$ is a generic function of $x$ that goes to zero as $|x|\to \infty$ and $\Phi$ is the CDF of the standard normal distribution.

Now what can we say about the distribution of the functions of $X_i$? For example, suppose that we have a Lipschitz function $f(X_1,\ldots,X_n) = \sqrt{\sum_{i=1}^n X_i^2}$, let $Z_i$ be standard Gaussian distributions such that $\mathbb{E}[Z_iZ_j]=\mathbb{E}[X_iX_j]$. What can we say about \begin{align}|P(f(X_1,\ldots,X_n)\leq x)- P(f(Z_1,\ldots,Z_n)\leq x)|\quad ?\hspace{20mm}(1)\end{align}

The continuous mapping theorem tells us that as $\rho\to 0$, the above quantity also converges to $0$. However I am interested in trying to find explicit convergence rates for $(1)$. Stein's method most likely plays a big role here, but most papers I've found are concerned with limiting distributions, eg. https://arxiv.org/pdf/1507.08688.pdf.

A similar question can be found in: Berry-Esseen bound in 2 dimensions for linear combinations, but I am hoping for a better answer/references if possible.