[proofplan]
We use the dual Bobkov-Gotze infimum-convolution characterization of the transport-entropy inequality and the Holley-Stroock bounded perturbation lemma for that dual formulation. The only measure-theoretic computation needed here is to identify the logarithmic oscillation of the density $d\mu/d\rho$. Since this density is $Z^{-1}e^{-V}$, its logarithmic oscillation is exactly $\operatorname{osc}(V)$, and the bounded perturbation lemma multiplies the transport constant by $e^{\operatorname{osc}(V)}$.
[/proofplan]
[step:Record the transport-entropy convention and the bounded perturbation input]
We use the following standard dual stability result.
(citing a result not yet in the wiki: Bobkov-Gotze infimum-convolution dual characterization of $T_p$ and the Holley-Stroock bounded perturbation lemma for transport inequalities.) Let $(X,d)$ be a Polish metric space, let $p\in\{1,2\}$, let $C\in(0,\infty)$, and let $\rho$ be a Borel probability measure on $(X,\mathcal B(X))$ satisfying $T_p(C)$. If $\alpha:X\to(0,\infty)$ is a bounded Borel measurable function satisfying
\begin{align*}
\int_X \alpha(x)\,d\rho(x)=1
\end{align*}
and if $\widetilde\rho$ is the Borel probability measure defined by
\begin{align*}
\frac{d\widetilde\rho}{d\rho}(x)=\alpha(x)
\end{align*}
for $\rho$-almost every $x\in X$, then $\widetilde\rho$ satisfies $T_p(Ce^{D})$, where
\begin{align*}
D:=\sup_{x\in X}\log\alpha(x)-\inf_{x\in X}\log\alpha(x).
\end{align*}
This is precisely the Holley-Stroock bounded perturbation principle in the normalization
\begin{align*}
W_p(\nu,\rho)^p\le C H(\nu\mid\rho).
\end{align*}
[/step]
[step:Verify that the perturbed density is a normalized bounded positive density]
Define the function
\begin{align*}
\alpha:X&\to(0,\infty)
\end{align*}
by
\begin{align*}
\alpha(x):=Z^{-1}e^{-V(x)}.
\end{align*}
Since $V$ is bounded, the functions $e^{-V}$ and $\alpha$ are bounded Borel measurable functions. Also $e^{-V(x)}>0$ for every $x\in X$, so
\begin{align*}
Z=\int_X e^{-V(x)}\,d\rho(x)
\end{align*}
satisfies $0<Z<\infty$. By the definition of $Z$,
\begin{align*}
\int_X \alpha(x)\,d\rho(x)=Z^{-1}\int_X e^{-V(x)}\,d\rho(x)=1.
\end{align*}
Thus $\mu=\alpha\rho$ is a Borel probability measure, and the bounded perturbation input applies once we compute the logarithmic oscillation of $\alpha$.
[guided]
The perturbation lemma is formulated for a normalized density $\alpha$ with respect to the original probability measure $\rho$, so we first verify exactly those hypotheses.
Define
\begin{align*}
\alpha:X&\to(0,\infty)
\end{align*}
by
\begin{align*}
\alpha(x):=Z^{-1}e^{-V(x)}.
\end{align*}
This is a Borel measurable function because $V:X\to\mathbb R$ is Borel measurable and the exponential function is continuous. Since $V$ is bounded, there are real numbers
\begin{align*}
a:=\inf_{x\in X}V(x)
\end{align*}
and
\begin{align*}
b:=\sup_{x\in X}V(x)
\end{align*}
with $-\infty<a\le b<\infty$. Therefore $e^{-V}$ is bounded above by $e^{-a}$ and bounded below by $e^{-b}$. Because $\rho$ is a probability measure, this gives
\begin{align*}
e^{-b}\le \int_X e^{-V(x)}\,d\rho(x)\le e^{-a}.
\end{align*}
Hence $0<Z<\infty$.
Now the normalization is not an additional assumption; it is built into the definition of $Z$. Indeed,
\begin{align*}
\int_X \alpha(x)\,d\rho(x)=\int_X Z^{-1}e^{-V(x)}\,d\rho(x).
\end{align*}
Since $Z^{-1}$ is a finite constant, linearity of the integral gives
\begin{align*}
\int_X \alpha(x)\,d\rho(x)=Z^{-1}\int_X e^{-V(x)}\,d\rho(x)=Z^{-1}Z=1.
\end{align*}
Thus $\alpha\rho$ is a Borel probability measure. By the definition in the theorem statement, this measure is exactly $\mu$.
[/guided]
[/step]
[step:Compute the logarithmic oscillation of the perturbation density]
For every $x\in X$,
\begin{align*}
\log\alpha(x)=-V(x)-\log Z.
\end{align*}
Since $-\log Z$ is constant in $x$, subtracting it does not change oscillation. Therefore
\begin{align*}
\sup_{x\in X}\log\alpha(x)-\inf_{x\in X}\log\alpha(x)=\sup_{x\in X}(-V(x))-\inf_{x\in X}(-V(x)).
\end{align*}
Using $\sup_X(-V)=-\inf_X V$ and $\inf_X(-V)=-\sup_X V$, we obtain
\begin{align*}
\sup_{x\in X}\log\alpha(x)-\inf_{x\in X}\log\alpha(x)=\sup_{x\in X}V(x)-\inf_{x\in X}V(x)=\operatorname{osc}(V).
\end{align*}
[/step]
[step:Apply the bounded perturbation principle to obtain the perturbed transport constant]
The hypotheses of the bounded perturbation input are satisfied with $\widetilde\rho=\mu$ and
\begin{align*}
D=\operatorname{osc}(V).
\end{align*}
Since $\rho$ satisfies $T_p(C)$ by assumption, the bounded perturbation principle gives that $\mu$ satisfies $T_p(Ce^{\operatorname{osc}(V)})$. Equivalently, for every Borel probability measure $\nu$ on $(X,\mathcal B(X))$,
\begin{align*}
W_p(\nu,\mu)^p\le Ce^{\operatorname{osc}(V)}H(\nu\mid\mu).
\end{align*}
This is the desired conclusion.
[/step]
