ABSTRACT Two particular challenges face type Ia supernovae (SNeIa) as probes of the expansion rate of the Universe. One is that they may not be fair tracers of the matter velocity field and the second is that their peculiar velocities distort the Hubble expansion. Although the latter has been estimated at ${\lesssim} 1.5~{{\ \rm per\ cent}}$ for $z>0.023$, this is based either on constrained linear or unconstrained (random) non-linear velocity simulations. In this paper, we address both challenges by incorporating a physical model for the locations of supernovae, and develop a Bayesian Hierarchical Model that accounts for non-linear peculiar velocities in our local Universe, inferred from a Bayesian analysis of the 2M++ spectroscopic galaxy catalogue. With simulated data, the model recovers the ground truth value of the Hubble constant $H_0$ in the presence of peculiar velocities including their correlated uncertainties arising from the Bayesian inference, opening up the potential of including lower redshift SNeIa to measure $H_0$. Ignoring peculiar velocities, the inferred $H_0$ increases minimally by ${\sim} 0.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$ in the range $0.023\ {<}\ z<0.046$. We conclude it is unlikely that the $H_0$ tension originates in unaccounted-for non-linear velocity dynamics.