I can prove that the decision problem of $Th(\langle \mathbb{N},= \rangle)$ is PSPACE-hard. However, the recursive algorithm to show it is in PSPACE would not work, since variables are unbounded and unrelated to the formula size.
Is there a more precise complexity classification for this theory?
Sentences in the empty language (= equality-only - I'm adopting the modern convention that equality is a logical predicate) have the finite model property in a very nice way: if $\varphi$ is a sentence in the empty language of length $n$, then the following are equivalent:
$\varphi$ is true in a pure set of size $n$.
$\varphi$ is true in every pure set of size $\ge n$.
This is proved by induction on formula complexity (really we just need to count variables occurring in $\varphi$ rather than its full length, but meh).
It's now easy to show that checking whether an equality-only sentence of length $n$ is satisfied in a pure set of size $k$ can be performed in space polynomial in $\max\{n,k\}$. This gives a $\mathsf{PSPACE}$ upper bound to your question (so we get $\mathsf{PSPACE}$-completeness).