The answer to your question is contained in the following local limit theorem for large deviations, due to [V. Petrov, Theorem 6][1]: 

> Suppose that $X,X_1,X_2,\dots$ are iid random variables such that $X$ only take values in the set $L:=\{a+kH\colon k\in\mathbb Z\}$ for some real $a$ and some real $h>0$, and suppose that the step $H$ is maximal with this property. Let $S_n:=X_1+\dots+X_n$, $R(h):=Ee^{hX}<\infty$ for all real $h>0$, $m(h):=R'(h)$, and $\sigma(h):=\sqrt{m'(h)}>0$, and $A_0:=\sup_{h>0} m(h)$. Then 
$$P(S_n=nx)=\frac H{\sigma(h_x)\sqrt{2\pi n}}\,\exp\{n\ln R(h_x)-nh_x x\}(1+O(1/n)), 
$$
where $x$ varies arbitrarily in any compact subinterval of the interval $(EX,A_0)$ so that $nx\in L$, and $h_x$ is the only root $h$ of the equation $m(h)=x$. 


[1]: https://epubs.siam.org/doi/10.1137/1110033