  We investigate the impact of different assumptions in the modeling of
one-loop galaxy bias on the recovery of cosmological parameters, as a follow up
of the analysis done in the first paper of the series at fixed cosmology. We
use three different synthetic galaxy samples whose clustering properties match
the ones of the CMASS and LOWZ catalogues of BOSS and the SDSS Main Galaxy
Sample. We investigate the relevance of allowing for either short range
non-locality or scale-dependent stochasticity by fitting the real-space galaxy
auto power spectrum or the combination of galaxy-galaxy and galaxy-matter power
spectrum. From a comparison among the goodness-of-fit ($\chi^2$), unbiasedness
of cosmological parameters (FoB), and figure-of-merit (FoM), we find that a
four-parameter model (linear, quadratic, cubic non-local bias, and constant
shot-noise) with fixed quadratic tidal bias provides a robust modelling choice
for the auto power spectrum of the three samples, up to $k_{\rm
max}=0.3\,h\,\mathrm{Mpc}^{-1}$ and for an effective volume of
$6\,h^{-3}\,\mathrm{Gpc}^3$. Instead, a joint analysis of the two observables
fails at larger scales, and a model extension with either higher derivatives or
scale-dependent shot-noise is necessary to reach a similar $k_{\rm max}$, with
the latter providing the most stable results. These findings are obtained with
three, either hybrid or perturbative, prescriptions for the matter power
spectrum, \texttt{RESPRESSO}, gRPT and EFT. In all cases, the inclusion of
scale-dependent shot-noise increases the range of validity of the model in
terms of FoB and $\chi^2$. Interestingly, these model extensions with
additional free parameters do not necessarily lead to an increase in the
maximally achievable FoM for the cosmological parameters
$\left(h,\,\Omega_ch^2,\,A_s\right)$, which are generally consistent to those
of the simpler model at smaller $k_{\rm max}$.
