Let $X_n$ and $Y_n$ be independent copies of two random variables $X$ and $Y$ with domain $\{-1,0,1\}$ for $n\in \mathbb{N}$. For a given $k\in \mathbb{N}$, I would like to find conditions on $X$ and $Y$ that guarantee that there is a $N\in \mathbb{N}$ such that for all $n>N$: $$P(\sum_{i=0}^{kn} X_i>0)>P(\sum_{i=0}^n Y_i>0). \tag{1}\label{eq1}$$

If the central limit theorem were exact in the sense that for large enough $n$, $F^X_n(x)=\Phi(x)=F^Y_n(x)$ where $F^X_n(x)$ and $F^Y_n(x)$ are the CDFs of the normalized sums of $X$ or $Y$ and $\Phi(x)$ is the CDF of the standard normal, condition \ref{eq1} would follow from $$\sqrt{k}\frac{\mathbb{E}( X_i)}{\sigma( X_i)}=\frac{\mathbb{E}(\sum_{i=0}^{k} X_i)}{\sigma(\sum_{i=0}^{k} X_i)}>\frac{\mathbb{E}( Y_i)}{\sigma(Y_i)}\tag{2}\label{eq2}.$$However, $P(\sum_{i=0}^{n} Y_i\leq 0)$ and $P(\sum_{i=0}^{kn} X_i\leq 0)$ seem to be of the order $e^{-n}$, while the Berry-Essen theorem only guarantees $|F^X_n(x)-\Phi(x)|\in O(n^{-0.5})$. Correspondingly, the normal approximation appears to be insufficient to draw conclusions about \ref{eq1}.

That said, condition \ref{eq2} appears to be remarkably accurate at predicting whether \ref{eq2} holds for large $n$ in a bunch of simulations I did. These simulations also suggested that in a large range of cases, $$F^{(\cdot)}_n(x)-\Phi(x)>0 \tag{3}\label{eq3}$$ at the relevant value of $x$ for both $X$ and $Y$. If that was true, one would only need to show that the error is eventually smaller for $X$ than for $Y$. I tried to prove \ref{eq3} using Edgeworth series, but the part of the residual for which I can control the sign seems to be dominated by the remaining error.

Are there any known conditions on when the CLT approximation error $F^{(\cdot)}_n(x)-\Phi(x)$ is positive for a given $x$ (either in the general case I presented or any subcase)?