Yes, every differential characteristic class is a stack morphism.
The point is that there exist universal differential characteristic classes. These are not easy to describe since they involve a notion of differential cohomology of classifying spaces. One way is to use Urs Schreiber's approach. At least in the case of degree four classes one can alternatively use the theory of multiplicative bundle gerbes.
In this context, I'd use the following working definition:
Definition: A universal degree four differential characteristic class on a Lie group $G$ is a multiplicative $U(1)$-bundle gerbe over $G$ with (multiplicative) connection of curvature $H$.
Here, $H$ is the "canonical" 3-form on $G$, it is defined using a bilinear symmetric linear form on the Lie algebra of $G$ - I think any such form is fine.
Multiplicative bundle gerbes have been introduced in the paper (1); they represent classes in $H^4(BG,\mathbb{Z})$. The notion of a multiplicative connection is subtle, I'd refer to Definition 1.3 of my paper (2). The point is that the "naive" definition is too strict and leaves essentially no space for examples. In particular, while a multiplicative gerbe over $G$ can be seen as a 2-gerbe over $BG$, a multiplicative connection is NOT a connection (in the ordinary sense) on this 2-gerbe.
Example 1: If $G=Spin(n)$, the basic gerbe $\mathcal{G}$ over $G$ carries a canonical connection of curvature $H$ and a canonical multiplicative structure. It is the universal differential half first Pontryagin class, $\frac{1}{2}\widehat{p_1}$. It underlies the definition of string connections I have proposed in my paper (3).
Example 2: If $G = SO(n)$, the bundle gerbe $\mathcal{G}$ descends together with its connection along the projection $Spin(n) \to SO(n)$, but its multiplicative structure does not descend. Instead, only the multiplicative bundle gerbe $\mathcal{G}^2 := \mathcal{G} \times \mathcal{G}$ descends together with its connection and its multiplciative structure. The descended gerbe over $SO(n)$ is the universal differential first Pontryagin class, $\widehat{p_1}$. Descent theory for multiplicative gerbes, together with obstructions is discussed in (4), see Table 1 at the end of the paper.
Now suppose that $\mathcal{G}$ is a universal differential characteristic class, $X$ is a smooth manifold and $P$ is a principal $G$-bundle with connection $A$ over $X$. The Chern-Simons 2-gerbe $\mathbb{CS}_P(\mathcal{G})$ is a $U(1)$-bundle 2-gerbe over $X$, see (1). A connection on $\mathbb{CS}_P(\mathcal{G})$ is constructed from the Chern-Simons 3-form $CS(A)$ and the multiplicative connection on $\mathcal{G}$; here the condition that the curvature of $\mathcal{G}$ is $H$ is important. This construction is described in detail in Section 3.2 of (2). It has the following properties:
Theorem:
if $\xi_P: M \to BG$ is a classifying map for $P$, then $[\mathbb{CS}_P(\mathcal{G})] = \xi_P^*[\mathcal{G}] \in H^4(M,\mathbb{Z})$.
a connection-preserving bundle morphism $P_1 \to P_2$ (covering some smooth map $X_1 \to X_2$ between base manifolds) induces a morphism $$ \mathbb{CS}_{P_1}(\mathcal{G}) \to \mathbb{CS}_{P_2}(\mathcal{G}) $$ between bundle 2-gerbes.
The first statement is Theorem 3.2.3 in (2). It means that on the level of the underlying (non-differential) characteristic class the construction is just pullback. The second statement follows by inspection of the definition of $\mathbb{CS}_P(\mathcal{G})$. It means precisely that we have a stack morphism $$ \mathbb{CS}(\mathcal{G}): Bun_G^{\nabla} \to 2\text{-}Grb_{U(1)}^{\nabla}. $$
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