Consider the boundary value problem for the compressible Navier-Stokes equation in a three-dimensional exterior domain, with a time-bounded external force and inhomogeneous boundary data for the temperature. We establish the existence of a time-bounded solution under smallness conditions on the external force and the boundary temperature. Furthermore, for sufficiently small initial perturbations, we prove global existence and time-decay estimates for solutions to the initial-boundary value problem for the perturbation system around this time-bounded solution.