[proofplan]
The proof is a cochain computation. The Chern-Weil transgression convention and the flatness of $A$ identify the coboundary of the singular integration cochain $I_{T P_0(A_0,A)}$ with $-I_{P_0(F_{A_0})}$. The chosen real cochain $r$ compares the real Chern-Weil cocycle with the chosen $\Lambda$-valued representative $\lambda$, while the cochain $\tau$ null-trivializes $\lambda$ over $\Lambda$. Adding the three coboundary identities gives a closed real cochain, whose reduction modulo $\Lambda$ gives the stated class and evaluation formula.
[/proofplan]
[step:Use transgression and flatness to compute the coboundary of the integration cochain]
Let
\begin{align*}
T:=T P_0(A_0,A)\in \Omega^{2k-1}(M)
\end{align*}
denote the chosen transgression form. By the Chern-Weil transgression formula, with the sign convention fixed in the statement,
\begin{align*}
dT=P_0(F_A)-P_0(F_{A_0}).
\end{align*}
Since $A$ is flat, $F_A=0$, and therefore $P_0(F_A)=0$. Hence
\begin{align*}
dT=-P_0(F_{A_0}).
\end{align*}
For the singular integration cochain $I_T\in C^{2k-1}(M;\mathbb R)$ on smooth singular chains, the form $T$ is smooth because it is the chosen Chern-Weil transgression form of smooth principal connections. Hence the singular Stokes formula applies and gives
\begin{align*}
\delta I_T=I_{dT}.
\end{align*}
Thus
\begin{align*}
\delta I_T=-I_{P_0(F_{A_0})}.
\end{align*}
[guided]
Let
\begin{align*}
T:=T P_0(A_0,A)\in \Omega^{2k-1}(M)
\end{align*}
be the transgression form chosen in the theorem statement. The relevant input is the Chern-Weil transgression identity, in the convention specified here:
\begin{align*}
dT=P_0(F_A)-P_0(F_{A_0}).
\end{align*}
This is exactly the convention needed to determine the sign of the secondary class.
Now use the hypothesis that $A$ is flat. Flatness means that its curvature form satisfies
\begin{align*}
F_A=0\in \Omega^2(M;\operatorname{ad}P).
\end{align*}
Because $P_0$ is $k$-linear, its Chern-Weil evaluation on the zero curvature form is zero:
\begin{align*}
P_0(F_A)=0.
\end{align*}
Substituting this into the transgression identity gives
\begin{align*}
dT=-P_0(F_{A_0}).
\end{align*}
We now pass from differential forms to singular cochains. For a smooth $j$-form $\omega\in\Omega^j(M)$, the singular integration cochain
\begin{align*}
I_\omega\in C^j(M;\mathbb R)
\end{align*}
is defined by integrating $\omega$ over smooth singular $j$-chains. The transgression form $T$ is smooth because it is built from the smooth principal connections $A_0$ and $A$ by the chosen Chern-Weil transgression construction. Therefore the singular Stokes formula applies to $T$ and says that singular integration commutes with the differentials:
\begin{align*}
\delta I_T=I_{dT}.
\end{align*}
Using the computation of $dT$ above, we obtain
\begin{align*}
\delta I_T=-I_{P_0(F_{A_0})}.
\end{align*}
This is the geometric part of the proof: the transgression form has precisely the coboundary needed to cancel the reference Chern-Weil cocycle.
[/guided]
[/step]
[step:Add the chosen trivialising cochains to obtain a closed real cochain]
View $\lambda$ and $\tau$ as real-valued cochains through the inclusion $\Lambda\subset\mathbb R$. Define
\begin{align*}
C:=I_T+r+\tau\in C^{2k-1}(M;\mathbb R).
\end{align*}
Using linearity of the singular coboundary operator and the defining identities for $r$ and $\tau$, we compute
\begin{align*}
\delta C=\delta I_T+\delta r+\delta\tau.
\end{align*}
Substituting the three coboundary identities gives
\begin{align*}
\delta C=-I_{P_0(F_{A_0})}+\bigl(I_{P_0(F_{A_0})}-\lambda\bigr)+\lambda.
\end{align*}
Therefore
\begin{align*}
\delta C=0.
\end{align*}
So $C$ is a closed real singular cochain. In particular, its reduction modulo $\Lambda$ is a closed cochain with coefficients in $\mathbb R/\Lambda$.
[/step]
[step:Reduce the closed cochain modulo $\Lambda$ to define the secondary class]
Let
\begin{align*}
\rho:\mathbb R\to \mathbb R/\Lambda
\end{align*}
be the quotient homomorphism. Applying $\rho$ coefficientwise defines a cochain map
\begin{align*}
\rho_\#:C^j(M;\mathbb R)\to C^j(M;\mathbb R/\Lambda)
\end{align*}
for every integer $j\ge 0$. Since $C$ is closed and $\rho_\#$ commutes with $\delta$, the cochain
\begin{align*}
\rho_\# C\in C^{2k-1}(M;\mathbb R/\Lambda)
\end{align*}
is closed. Define
\begin{align*}
\operatorname{CS}_{P_0}(A;A_0,r,\tau):=[\rho_\# C]\in H^{2k-1}(M;\mathbb R/\Lambda).
\end{align*}
This is the asserted secondary Chern-Simons class.
[/step]
[step:Evaluate the representing cocycle on a closed singular cycle]
Let
\begin{align*}
Z\in Z_{2k-1}(M;\mathbb Z)
\end{align*}
be a smooth singular cycle. By definition of cohomology evaluation, the class represented by $\rho_\# C$ evaluates on $[Z]$ as
\begin{align*}
\operatorname{CS}_{P_0}(A;A_0,r,\tau)([Z])=(\rho_\# C)(Z).
\end{align*}
Using the definition of $C$, this equals
\begin{align*}
(\rho_\# C)(Z)=\left[I_T(Z)+r(Z)+\tau(Z)\right]\in \mathbb R/\Lambda.
\end{align*}
Since $I_T$ is the singular integration cochain of the transgression form $T=T P_0(A_0,A)$, this gives
\begin{align*}
\operatorname{CS}_{P_0}(A;A_0,r,\tau)([Z])=\left[I_{T P_0(A_0,A)}(Z)+r(Z)+\tau(Z)\right]\in \mathbb R/\Lambda.
\end{align*}
This is the stated evaluation formula.
[/step]
