Compatibility of Drinfeld presentations for split affine Kac-Moody quantum symmetric pairs

Let $(\mathbf{U},{\mathbf{U}}^{\mathrm{ı}})$ 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 ${\mathrm{\Theta }}_i(z)$ 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.