Representations of twisted Yangians of types B, C, D: II

We continue the study of finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types B, C and D, with focus on those of types BI, CII and DI. After establishing that, for all twisted Yangians of these types, the highest weight of such a module necessarily satisfies a certain set of relations, we classify the finite-dimensional irreducible representations of twisted Yangians for the pairs $(\mathfrak{so}_N,\mathfrak{so}_{N-2} \oplus \mathfrak{so}_2)$ and $(\mathfrak{so}_{2n+1},\mathfrak{so}_{2n})$.