$\newcommand\ep\epsilon$Let $Z$ denote a standard normal random variable. The condition $|X_i|\le1$ implies that $A_3:=\sum_{i=1}^n E|X_i|^3\le\sum_{i=1}^n E|X_i|^2 =s_n^2$. Also note that $P(1+|S_n|\le\epsilon s_n)=0$ unless $1\le\ep s_n$.
Therefore and in view of the Berry--Esseen inequality with Shevtsova's constant factor $0.5600$, for real $\ep\ge0$, $$\begin{align} &P(1+|S_n|\le\epsilon s_n) \\ &\le1(1\le\ep s_n)P(|S_n|\le\epsilon s_n) \\ &\le 1(1\le\ep s_n) \Big(P(|Z|\le\ep)+0.5600\,\frac{A_3}{s_n^3}\Big) \\ &\le 1(1\le\ep s_n)\Big(\frac2{\sqrt{2\pi}}\,\ep+\frac{0.5600}{s_n}\Big) \\ &\le\Big(\frac2{\sqrt{2\pi}}+0.5600\Big)\ep \le1.4\ep. \end{align}$$ So, $P(1+|S_n|\le\epsilon s_n)\to0$ uniformly in $n$ as $\ep\downarrow0$. Thus, $P(1+|S_n|>\epsilon s_n)\to1$ uniformly in $n$ as $\ep\downarrow0$.