On the $8$-rank of narrow class groups of $\mathbb{Q}(\sqrt{-4pq})$, $\mathbb{Q}(\sqrt{-8pq})$, and $\mathbb{Q}(\sqrt{8pq})$

Let $d\in \{-4,-8,8\}$. We study the $8$-part of the narrow class group in the thin families of quadratic number fields of the form $\mathbb{Q}(\sqrt{dpq})$, where $p\equiv q\equiv 14$ are prime numbers, and we prove new lower bounds for the proportion of narrow class groups in these families that have an element of order $8$. In the course of our proof, we prove a general double-oscillation estimate for the quadratic residue symbol in quadratic number fields.