In this paper, we show a large deviation principle for certain sequences of static Schrödinger bridges, typically motivated by a scale-parameter decreasing towards zero, extending existing large deviation results to cover a wider range of reference processes. Our results provide a theoretical foundation for studying convergence of such Schrödinger bridges to their limiting optimal transport plans. Within generative modeling, Schrödinger bridges, or entropic optimal transport problems, constitute a prominent class of methods, in part because of their computational feasibility in high-dimensional settings. Recently, Bernton et al. established a large deviation principle, in the small-noise limit, for fixed-cost entropic optimal transport problems. In this paper, we address an open problem posed by Bernton et al. and extend their results to hold for Schrödinger bridges associated with certain sequences of more general reference measures with enough regularity in a similar small-noise limit. These can be viewed as sequences of entropic optimal transport plans with non-fixed cost functions. Using a detailed analysis of the associated Skorokhod maps and transition densities, we show that the new large deviation results cover Schrödinger bridges where the reference process is a reflected diffusion on bounded convex domains, corresponding to recently introduced model choices in the generative modeling literature.