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

Let $(U,U^{ı})$ be a split affine quantum symmetric pair of type $B_n^{\left(1\right)},C_n^{\left(1\right)}$ or  $D_n^{\left(1\right)}$ . We prove factorization and coproduct formulae for the Drinfeld-Cartan operators ${\Theta }_i\left(z\right)$ in the Lu-Wang Drinfeld-type presentation, generalizing the type $A_n^{\left(1\right)}$ result from Przeździecki (arXiv:2311.13705). 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.