  The implementation of star formation and stellar feedback in cosmological
simulations plays a critical role in shaping galaxy properties. In the first
paper of the series, we presented a new method to model star formation as a
collection of star clusters. In this paper, we improve the algorithm by
eliminating accretion gaps, boosting momentum feedback, and introducing a
subgrid initial bound fraction, $f_i$, that distinguishes cluster mass from
stellar particle mass. We perform a suite of simulations with different star
formation efficiency per freefall time $\epsilon_{\rm ff}$ and supernova
momentum feedback intensity $f_{\rm boost}$. We find that the star formation
history of a Milky Way-sized galaxy is sensitive to $f_{\rm boost}$, which
allows us to constrain its value, $f_{\rm boost}\approx5$, in the current
simulation setup. Changing $\epsilon_{\rm ff}$ from a few percent to 200\% has
little effect on global galaxy properties. However, on smaller scales, the
properties of star clusters are very sensitive to $\epsilon_{\rm ff}$. We find
that $f_i$ increases with $\epsilon_{\rm ff}$ and cluster mass. Through the
dependence on $f_i$, the shape of the cluster initial mass function varies
strongly with $\epsilon_{\rm ff}$. The fraction of clustered star formation and
maximum cluster mass increase with the star formation rate surface density,
with the normalization of both relations dependent on $\epsilon_{\rm ff}$. The
cluster formation timescale systematically decreases with increasing
$\epsilon_{\rm ff}$. Local variations in the gas accretion history lead to a
0.25~dex scatter for the integral cluster formation efficiency. Joint
constraints from all the observables prefer the runs that produce a median
integral efficiency of 16\%.
