Find the $k$-th lexicographically smallest hamiltonian circuit
Counting the number of Hamiltonian circuits in a graph is $\mathsf{\#P}$-hard (see for example this answer). Your problem is even harder, since we can use your problem together with binary search to count the number of Hamiltonian circuits in a given graph.

As an aside, let me mention that listing things in lexicographical order might be harder than listing them in any order. A case in point is the complexity of finding a [maximal independent set], which is an independent set which cannot be enlarged (but not necessarily one which has maximum size). A maximal independent set can easily be found by going over the vertices in some arbitrary order, putting the problem in the complexity class $\mathsf{P}$. This algorithm can be improved, and in fact a maximal independent set can be found in $\mathsf{NC^2}$. In contrast, finding the lexicographically first maximal independent set (that is, deciding whether a given vertex belongs to this set) is $\mathsf{P}$-complete.