We analyze the \emph{equilibrium fluctuations} of a Hamiltonian chain of oscillators on \(\mathbb{Z}\) with an exponential potential, perturbed by a conservative, symmetric noise. Under the canonical \emph{diffusive scaling} \(t \mapsto t n^2\) and an interaction strength tuned by \(n^{-1/2}\), the fluctuation field is known to converge to the \emph{energy solution} of the stochastic Burgers equation (SBE) on the torus~\cite{ABGS22}. We introduce a \emph{coupled moving heat bath} of strength \(n^{-δ}\) acting on the particle system. We prove that for \(δ\leq 1\) (the \emph{strong-coupling regime}), the equilibrium fluctuation field converges to the \emph{energy solution of the SBE with a Dirichlet boundary condition at zero}. We provide two distinct analytical characterizations of these boundary solutions, corresponding to different spaces of test functions. Conversely, for \(δ> 1\) (the \emph{weak-coupling regime}), the heat bath becomes irrelevant in the scaling limit: the fluctuations converge to the standard SBE on the full line without any boundary condition, reproducing the full-line result of~\cite{GJ14}. Our analysis thus reveals a sharp \emph{critical scaling} in the coupling strength \(δ\), which dictates the emergence -- or absence -- of a macroscopic boundary condition from the microscopic perturbation.