[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 $\alpha$.[/step]

[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: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]