Detection and quantification of greenhouse-gas emissions is important for both compliance and environment conservation. However, despite several decades of active research, it remains predominantly an open problem, largely due to model errors and assumptions that appear at each stage of the inversion processing chain. In 2015, a controlled-release experiment headed by Geoscience Australia was carried out at the Ginninderra Controlled Release Facility, and a variety of instruments and methods were employed for quantifying the release rates of methane and carbon dioxide from a point source. This paper proposes a fully Bayesian approach to atmospheric tomography for inferring the methane emission rate of this point source using data collected during the experiment from both point-and path-sampling instruments. The Bayesian framework is designed to account for uncertainty in the parameterisations of measurements, the meteorological data, and the atmospheric model itself when performing inversion using Markov chain Monte Carlo (MCMC). We apply our framework to all instrument groups using measurements from two release-rate periods. We show that the inversion framework is robust to instrument type and meteorological conditions. From all the inversions we conducted across the different instrument groups and release-rate periods, our worst-case median emission rate estimate was within 36 % of the true emission rate. Further, in the worst case, the closest limit of the 95 % credible interval to the true emission rate was within 11 % of this true value.