  Accurate mock galaxy catalogues are crucial to validate analysis pipelines
used to constrain dark energy models. We present a fast HOD-fitting method
which we apply to the AbacusSummit simulations to create a set of mock
catalogues for the DESI Bright Galaxy Survey, which contain r-band magnitudes
and g-r colours. The halo tabulation method fits HODs for different absolute
magnitude threshold samples simultaneously, preventing unphysical HOD crossing
between samples. We validate the HOD fitting procedure by fitting to real-space
clustering measurements and galaxy number densities from the MXXL BGS mock,
which was tuned to the SDSS and GAMA surveys. The best-fitting clustering
measurements and number densities are mostly within the assumed errors, but the
clustering for the faint samples is low on large scales. The best-fitting HOD
parameters are robust when fitting to simulations with different realisations
of the initial conditions. When varying the cosmology, trends are seen as a
function of each cosmological parameter. We use the best-fitting HOD parameters
to create cubic box and cut sky mocks from the AbacusSummit simulations, in a
range of cosmologies. As an illustration, we compare the Mr<-20 sample of
galaxies in the mock with BGS measurements from the DESI one-percent survey. We
find good agreement in the number densities, and the projected correlation
function is reasonable, with differences that can be improved in the future by
fitting directly to BGS clustering measurements. The cubic box and cut-sky
mocks in different cosmologies are made publicly available.
