This combines comments of myself and Lee Mosher.
Exercise 7.15 of Miller and Sturmfels combinatorial commutative algebra book proves the following. Let $N=C_U\cap G_U$. Then $N$ is a finitely generated semigroup and there exists according to exercise 7.15 an element $a$ of $S_U$ with $a+N\subseteq S_U$. Now take $R>|a|$. If $B_R(v)\subseteq C_U$ then $v−a\in C_U\cap G_U=N$ so $v\in a+N\subseteq S_U$.