Inspired by the confirmed detection of a short gamma-ray burst (GRB) in association with a gravitational wave signal, we present the first Bayesian Fermi-Gamma-ray Burst Monitor (GBM) short GRB spectral catalogue. Both peak flux and time-resolved spectral results are presented. Data are analysed with the proper Poisson likelihood allowing us to provide statistically reliable results even for spectra with few counts. All fits are validated with posterior predictive checks. We find that nearly all spectra can be modelled with a cut-off power law. Additionally, we release the full posterior distributions and reduced data from our sample. Following our previous study, we introduce three variability classes based on the observed light-curve structure.