We apply BayeSN, our new hierarchical Bayesian model for the SEDs of Type Ia supernovae (SNe Ia), to analyse the griz light curves of 157 nearby SNe Ia (0.015< z< 0.08) from the public Foundation DR1 dataset. We train a new version of BayeSN, continuous from 0.35-0.95 microns, which we use to model the properties of SNe Ia in the rest-frame z-band, study the properties of dust in their host galaxies, and construct a Hubble diagram of SN Ia distances. Our Hubble diagram has a low total RMS of 0.13 mag using BayeSN, compared to 0.16 mag using a conventional model (SALT2). We test the consistency of dust laws between low- and high-mass host galaxies by using our model to fit the full time- and wavelength-dependent SEDs of SNe Ia up to moderate reddening (peak apparent B-V< 0.3). Modelling population distributions of the dust law R_V in low- and high-mass hosts, we find that both subsamples are highly consistent with the full sample's population mean of 2.70+/-0.25, with a 95% upper bound on the population std. dev. of 0.61. We find that simultaneous fitting of host-mass-dependent dust properties within our hierarchical model does not account for the conventional mass step.