In this paper, for the current algebra associated with the Lie algebra $\lie{so}_{2n}(\mathbb{C})$, we connect particular CV-modules, indexed by pairs of integral weights $(\lambda,\mu)$ that satisfy particular conditions, with generalized Demazure modules which were introduced by [Naoi]. In particular, we show an isomorphism of a CV-module $V(\xi(\lambda,\mu))$ and the module generated by the tensor product of generating vectors $v_\lambda$, $v_\mu$ for the local Weyl modules $W_{\loc}(\lambda)$ and $W_{\loc}(\mu)$ respectively, i.e. $V(\xi(\lambda,\mu))\cong \langle v_\lambda \otimes v_\mu\rangle \subset W_{\loc}(\lambda) \otimes W_{\loc}(\mu)$. To complete this proof, we construct short exact sequences of CV-modules. Moreover, through this construction, we obtain a Demazure character formula for the CV-module $V(\xi(\lambda,\mu))$.