Let me clarify the role of the center in my question. The braided functor to the center is needed to extend the ordinary monoidal category structure on $\mathcal{C}$ to a monoidal $\mathcal{V}$-category structure.
A strong braided monoidal functor $z\colon \mathcal{V}\rightarrow z(\mathcal{C})$ is the same as a strong monoidal functor $z\colon \mathcal{V}\rightarrow \mathcal{C}$ together with natural isomorphisms $\zeta(X,Y) : z(X)\otimes Y \cong Y\otimes z(X)$ satisfying some coherence laws. We need these isomorphisms to define the $\mathcal{V}$-enrichment of the tensor product in $\mathcal{C}$ \mathcal{C}$: $$\otimesHom_{\mathcal{C}}(W,X)\otimes $\otimes : Hom_{\mathcal{C}}(W,X)\otimes Hom_{\mathcal{C}}(Y,Z)\longrightarrow Hom_{\mathcal{C}}(W\otimes Y, X\otimes Z)$$Z).$$ This morphism must be the adjoint of:
$\qquad z(Hom_{\mathcal{C}}(W,X)\otimes Hom_{\mathcal{C}}(Y,Z))\otimes W\otimes Y$
$\cong z(Hom_{\mathcal{C}}(W,X))\otimes z(Hom_{\mathcal{C}}(Y,Z))\otimes W\otimes Y$
$\stackrel{\zeta}\cong z(Hom_{\mathcal{C}}(W,X))\otimes W\otimes z(Hom_{\mathcal{C}}(Y,Z))\otimes Y\stackrel{ev \otimes ev}\longrightarrow X \otimes Z.$
