KLR algebras and the branching rule II: the categorical Gelfand-Tsetlin basis for the classical Lie algebras

We construct functors categorifying the branching rules for $U_q(\mathfrak{g})$ for $\mathfrak{g}$ of type $B_n$, $C_n$, and $D_n$ for the embeddings $s{o}_{2n+1}\supset s{o}_{2n-1}$, $s{p}_{2n}\supset s{p}_{2n-2}$, and $s{o}_{2n}\supset s{o}_{2n-2}$. We give the corresponding categorical Gelfand-Tsetlin basis.