Query: Given a graph, is edge x in an optimal TSP tour?
The complexity class $\Sigma_2^P$, which is the second level of the polynomial hierarchy, consists of all languages $L$ for which there exists a polytime predicate $f$ and a polynomial $p$ such that
$$
x \in L \Longleftrightarrow \exists y \forall z \, f(x,y,z), \text{ where } |y|,|z| \leq p(|x|).
$$
This is the natural location for your problem:


$x$ is a pair $(G,e)$, where $G$ is a graph and $e$ is an edge of $G$.
$y$ is a TSP tour.
$z$ is a TSP tour.
$f(x,y,z)$ holds if (i) $y,z$ are TSP tours of $G$; (ii) $e \in y$; and (iii) $w(y) \leq w(z)$.


More generally, $\Sigma_k^P$ is defined similarly with $k$ quantifier alternations, and $\Pi_k^P$ is defined the same but starting with a universal rather than an existential quantifier. For example, $\Sigma_1^P$ is NP and $\Pi_1^P$ is coNP. We also define $\Delta_k^P = \Sigma_k^p \cap \Pi_k^P$, generalizing the class NP∩coNP.

We define $\Sigma_k^P$-hard and $\Sigma_k^P$-complete in analog to NP-hard and NP-complete, using polynomial time reductions. The obvious question to ask in your case is whether your language is $\Sigma_2^P$-complete.