$\newcommand\ep\epsilon$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 $0.5600$][1], 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 $\ep\downarrow0$. 


  [1]: https://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem#Non-identically_distributed_summands