[proofplan]
The proof uses only the naturality of the fixed Cheeger-Simons characteristic class. A gauge transformation is a bundle automorphism covering the identity on $M$, and the gauge-transformed connection is defined so that this automorphism is connection-preserving. Naturality then identifies the two differential characters on $M$; since the base map is $\operatorname{id}_M$, evaluating on the fundamental cycle gives the same element of $\mathbb R/\mathbb Z$. This proves constancy on gauge orbits and therefore gives the descended map.
[/proofplan]
[step:Identify a gauge transformation as a connection-preserving bundle isomorphism]
Let $u\in \mathcal G(E)$ be a gauge transformation. Thus $u:E\to E$ is a smooth principal $G$-bundle automorphism satisfying
\begin{align*}
\pi\circ u=\operatorname{id}_M\circ \pi,
\end{align*}
where $\pi:E\to M$ is the bundle projection.
Let $A\in \mathcal A_{\mathrm{flat}}(E)$. Denote by $u\cdot A$ the gauge-transformed connection. By the definition of the gauge action on connections, the bundle automorphism $u$ is an isomorphism of principal bundles with connection
\begin{align*}
u:(E,A)\longrightarrow (E,u\cdot A)
\end{align*}
covering the base diffeomorphism $\operatorname{id}_M:M\to M$.
Because curvature is carried by connection-preserving bundle isomorphisms, $u\cdot A$ is flat whenever $A$ is flat. Hence the gauge action restricts to an action of $\mathcal G(E)$ on $\mathcal A_{\mathrm{flat}}(E)$.
[/step]
[step:Apply naturality of the Cheeger-Simons characteristic class]
The fixed class $\widehat c_P$ is natural for isomorphisms of principal bundles with connection. Therefore, applied to the connection-preserving bundle isomorphism
\begin{align*}
u:(E,A)\longrightarrow (E,u\cdot A)
\end{align*}
over $\operatorname{id}_M$, naturality gives
\begin{align*}
\widehat c_P(E,A)=\operatorname{id}_M^*\widehat c_P(E,u\cdot A)
\end{align*}
as elements of $\widehat H^{2k}(M;\mathbb Z)$.
Since pullback by the identity map is the identity operation on differential characters, this reduces to
\begin{align*}
\widehat c_P(E,A)=\widehat c_P(E,u\cdot A).
\end{align*}
[guided]
We now use the only structural property of the chosen Cheeger-Simons refinement that is needed for gauge invariance: naturality. The gauge transformation $u:E\to E$ is not merely a map of total spaces; it is a principal $G$-bundle automorphism covering the identity map on the base. The connection $u\cdot A$ is defined precisely so that
\begin{align*}
u:(E,A)\longrightarrow (E,u\cdot A)
\end{align*}
is an isomorphism of principal bundles equipped with connection.
Naturality of the fixed differential characteristic class says that if an isomorphism of bundles with connection covers a smooth map $f:X\to Y$, then the differential character assigned to the source is the pullback by $f$ of the differential character assigned to the target. In the present case the source and target base are both $M$, and the covering map is
\begin{align*}
\operatorname{id}_M:M\to M.
\end{align*}
Thus naturality gives
\begin{align*}
\widehat c_P(E,A)=\operatorname{id}_M^*\widehat c_P(E,u\cdot A).
\end{align*}
The pullback of a differential character by $\operatorname{id}_M$ is the same differential character, because $\operatorname{id}_M$ leaves every smooth singular cycle in $M$ unchanged. Hence
\begin{align*}
\widehat c_P(E,A)=\widehat c_P(E,u\cdot A)
\end{align*}
in $\widehat H^{2k}(M;\mathbb Z)$.
[/guided]
[/step]
[step:Evaluate the equal differential characters on the fundamental cycle]
Since $M$ is closed and oriented of dimension $2k-1$, its orientation determines a fundamental cycle representing
\begin{align*}
[M]\in H_{2k-1}(M;\mathbb Z).
\end{align*}
A differential character of degree $2k$ is evaluated on smooth singular $(2k-1)$-cycles, with values in $\mathbb R/\mathbb Z$. Evaluating the equality
\begin{align*}
\widehat c_P(E,A)=\widehat c_P(E,u\cdot A)
\end{align*}
on the fundamental cycle $[M]$ gives
\begin{align*}
\widehat c_P(E,A)([M])=\widehat c_P(E,u\cdot A)([M])
\end{align*}
in $\mathbb R/\mathbb Z$.
[/step]
[step:Descend the invariant to the gauge quotient]
The preceding step proves that for every $A\in \mathcal A_{\mathrm{flat}}(E)$ and every $u\in \mathcal G(E)$,
\begin{align*}
\operatorname{CS}_{P,E}(u\cdot A)=\operatorname{CS}_{P,E}(A).
\end{align*}
Thus $\operatorname{CS}_{P,E}$ is constant on each $\mathcal G(E)$-orbit in $\mathcal A_{\mathrm{flat}}(E)$.
Define
\begin{align*}
\overline{\operatorname{CS}}_{P,E}: \mathcal A_{\mathrm{flat}}(E)/\mathcal G(E) &\to \mathbb R/\mathbb Z
\end{align*}
\begin{align*}
[A] &\mapsto \widehat c_P(E,A)([M]).
\end{align*}
If $[A]=[A']$ in $\mathcal A_{\mathrm{flat}}(E)/\mathcal G(E)$, then there exists $u\in \mathcal G(E)$ such that $A'=u\cdot A$, and the equality above gives
\begin{align*}
\widehat c_P(E,A')([M])=\widehat c_P(E,A)([M]).
\end{align*}
Therefore the displayed formula is independent of the chosen representative $A$ of the gauge-equivalence class. This proves that the Chern-Simons invariant modulo integers descends to the stated well-defined map.
[/step]
