$x \in (a,b) \subset \Bbb N$ as a way to say "$x$ is any natural number in the interval $a < x < b$."
I like this expression better than $x \in \Bbb N, x \in (a,b)$, but I'm not sure if it's allowed, since by definition, an interval is a subset of the reals. However, a set that only contains positive integers would still be a subset of the reals, but it would also be a subset of the naturals, no?
It is not correct, as you mentioned. By definition, $(a,b)=\{x\in \mathbb R : a<x<b\}$, which is not a subset of $\mathbb N$.
You can do what you want by writing
$$x\in(a,b)\cap \mathbb N,$$
that is, the intersection of the interval with the natural numbers.