$\newcommand\dee{\Delta_{\mathrm{TV}}}\newcommand{\vpi}{\varphi}$Let $f_n$ denote the pdf of 
\begin{equation*}
	S_n:=\sqrt{\frac3n} \sum_{i=1}^n X_i
\end{equation*}
and let $\vpi$ denote the standard normal pdf. Then 
\begin{equation*}
	\dee(D_n,N(0,1))=\int|f_n-\vpi|\le I_{n1}+I_{n2}+I_{n3}, \tag{1}\label{1}
\end{equation*}
where 
\begin{equation*}
	I_{n1}:=\int_{|x|\le\sqrt{3n}}|f_n(x)-\vpi(x)|\,dx,
\end{equation*}
\begin{equation*}
	I_{n2}:=\int_{|x|>\sqrt{3n}}f_n(x)\,dx,
\end{equation*}
\begin{equation*}
	I_{n3}:=\int_{|x|>\sqrt{3n}}\vpi(x)\,dx. 
\end{equation*}

We bound $I_{n1}$ using the asymptotic expansion in the central limit theorem given (say) by Theorem 7 (with $k=5$ and $l=1$) on p. 175 in the book by [Petrov][1]. Noting also Lemma 10 on p. 173 and the expressions for $Q_{kn}(x)$ on p. 138 of the same book, as well as the fact that $EX_i=EX_i^3=0$ for all $i$, we see that 
\begin{equation*}
	f_n(x)=\vpi(x)\Big(1+\frac{P_2(x)}n\Big)+O\Big(\frac1{n^{3/2}}\Big) 
\end{equation*}
uniformly in all real $x$, where $P_2$ is a certain polynomial. It follows that $I_{n1}=O(1/n)$. Also, $I_{n2}=0$, since $|S_n|\le\sqrt{\frac3n}\,n=\sqrt{3n}$. Finally, it is easy to see that $I_{n3}=O(1/n)$. 

Thus, by \eqref{1},  
 \begin{equation*}
	\dee(D_n,N(0,1))=O(1/n), 
\end{equation*}
as desired. 

---

In fact, $P_2(x)=\frac1{20}(3-6x^2+x^4)$, and hence the above reasoning shows that 
 \begin{equation*}
	\dee(D_n,N(0,1))\sim c/n, 
\end{equation*}
where 
\begin{equation}
	c:=\int\vpi|P_2|=
	\frac{e^{-3/2-\sqrt{3/2}}}{5\sqrt\pi}  
	\Big(e^{\sqrt{6}} \sqrt{9-3\sqrt{6}}+\sqrt{9+3\sqrt{6}}\,\Big) \\ 
   =0.140030\ldots. 
\end{equation}

Here is the graph $\big\{\big(n,\frac nc\,\dee(D_n,N(0,1))\big)\colon n\in\{3,\dots,15\}\big\}$: 

[![enter image description here][2]][2]


  [1]: https://link.springer.com/book/10.1007/978-3-642-65809-9
  [2]: https://i.sstatic.net/g7J5C.png