The emergence of precision gravity simulators in quantum and fluid systems is opening new avenues for probing curved-spacetime physics and black-hole phenomenology under controlled laboratory conditions. In parallel, advances in understanding how fundamental physics can be probed in the spectral signatures of black holes and exotic compact objects motivate the development of modern spectroscopic techniques within analogue-gravity experiments. In this work, we model the spectral properties of analogue black holes sourced by broadband stochastic noise, a crucial aspect in realistic experiments that poses substantial challenges for established data-analysis techniques. Using simulation-based inference, we demonstrate on simulated spectra that the physical parameters encoded in noisy spectra can be reliably extracted. The inference retains predictive power even under out-of-distribution red-noise contamination. These results establish simulation-based inference as a promising route to spectroscopy in laboratory gravity simulators.