Let $E$ be a CM elliptic curve defined over a quadratic imaginary field $K$ with maximal order i.e., $\mathrm{End}_K(E)\cong \mathcal{O}=\mathcal{O}_K$. Let $\mathfrak{p}$ be a prime of $K$ such that the map $\mathcal{O}^\times \to(\mathcal{O}/\mathfrak{p})^\times$ is not surjective. With this situation, I tried to prove $E[\mathfrak{p}]\not \subset E(K)$.

My idea is as follows; If $E[\mathfrak{p}]\subset E(K)$, then we can consider $(\mathcal{O}/\mathfrak{p})^\times$ as the set of isomorphisms modulo an equivalence relation `$\sim$' defined by for $f,g\in \mathcal{O}$ we say $f\sim g$ when they coincide on $E[\mathfrak{p}]$. Then it seems contradict to the assumption $\mathcal{O}^\times \to(\mathcal{O}/\mathfrak{p})^\times$ is not surjective. Is it plausible? If it is, how to expand details of proof?