Notation for asymptotic bounds on both sides
I'm also not aware of any specific notation for the concept you want. In the example theorem statement in the question, there's nothing wrong with what you already have ("$\epsilon$ such that $\epsilon=o(\cdots)$ and $\epsilon=\omega(\cdots)$"). In other situations, it might be more convenient to say something like, "For any $\epsilon\in o(\cdots)\cap\omega(\cdots)$".