  Let $(\mathbf{U}, \mathbf{U}^\imath)$ be a split affine quantum symmetric
pair of type $\mathsf{B}_n^{(1)}, \mathsf{C}_n^{(1)}$ or $\mathsf{D}_n^{(1)}$.
We prove factorization and coproduct formulae for the Drinfeld-Cartan operators
$\Theta_i(z)$ in the Lu-Wang Drinfeld-type presentation, generalizing the type
$\mathsf{A}_n^{(1)}$ result from [Prz23]. As an application, we show that a
boundary analogue of the $q$-character map, defined via the spectra of these
operators, is compatible with the usual $q$-character map. As an auxiliary
result, we also produce explicit reduced expressions for the fundamental
weights in the extended affine Weyl groups of classical types.
