Let $k$ be a finite field. Let $K$ a finite extension of $k(t)$. Let $S$ be a finite set of places of $K$. Let $$K_S = \prod_{v\in S} K_v$$ where $K_v$ is the completion of $K$ at $v$. For $v\in S$, let $\mathfrak{m}_v\subset\mathcal{O}_{K_v}$ be the maximal ideal of the valuation ring $\mathcal{O}_{K_v}$ of $K_v$. Let $$\mathcal{O}_{K,S} = \{\alpha\in K : v(\alpha)\geq 0, \forall v\not\in S\}\subset K$$ be the $S$-integers and embed $\mathcal{O}_{K,S}$ into $K_S$ diagonally. Let $\mathfrak{m}_S = \prod_{v\in S}\mathfrak{m}_v$. By the product formula, $\mathcal{O}_{K,S}\cap \mathfrak{m}_S = \{0\}$, so we get an embedding $$\mathfrak{m}_S\hookrightarrow \frac{K_S}{\mathcal{O}_{K,S}}.$$ My question is, is this embedding surjective?
The reason why I stated weak approximation in the title is because this question can be restated as follows: Given $\alpha = (\alpha_v)_{v\in S}\in K_S$, does there exists $\beta\in \mathcal{O}_{K,S}$ such that for all $v\in S$, $\alpha_v - \beta\in \mathfrak{m}_v$? If $\mathcal{O}_{K,S}$ is replaced by $K$ in this question, then the answer is yes by weak approximation (and we can replace the $\mathfrak{m}_v$'s by any power of the $\mathfrak{m}_v$'s).
My intuition says the answer to the question is yes, but after trying for a while I cannot prove it or find a reference. The most trivial example is the fact that $$k((t^{-1})) = k[t] \oplus t^{-1}k[[t^{-1}]]$$ where the direct sum is as abelian groups. In this example, $K = k(t)$ and $S = \{v\}$ where $v$ is the valuation with uniformizer $t^{-1}$. Then $\mathcal{O}_{K,S} = k[t]$ and $K_S = k((t^{-1}))$.
Any hint, answer, or reference would be appreciated!
$\textbf{Edit}$: By Felipe's comment, the answer to the original question is no, and a necessary condition for $\beta$ to exist is that $$\sum_{v\in S}res_v(\alpha_v\omega) = 0$$ for all differentials $\omega$ of $K$. My two new questions are, is there a sufficient condition for such a $\beta$ to exist? And is there a description or meaning/significance to the quotient $\displaystyle \frac{K_S}{\mathcal{O}_{K,S} + \mathfrak{m}_S}$?
