  In this work, we investigate an alternative channel for the formation of
fast-spinning black hole-neutron star (BHNS) binaries, in which super-Eddington
accretion is expected to occur in accreting BHs during the stable mass transfer
phase within BH-stripped helium (BH--He-rich) star binary systems. We evolve
intensive \texttt{MESA} grids of close-orbit BH--He-rich star systems to
systematically explore the projected aligned spins of BHs in BHNS binaries, as
well as the impact of different accretion limits on the tidal disruption
probability and electromagnetic (EM) signature of BHNS mergers. Most of the BHs
in BHNS mergers cannot be effectively spun up through accretion, if the
accretion rate is limited to $\lesssim10\,\dot{M}_{\rm Edd}$, where
$\dot{M}_{\rm Edd}$ is the standard Eddington accretion limit. In order to
reach high spins (e.g., $\chi_{\rm BH} \gtrsim 0.5$), the BHs are required to
be born less massive (e.g., $\lesssim3.0\,M_\odot$) in binary systems with
initial periods of $\lesssim0.2-0.3\,{\rm days}$ and accrete material at
$\sim100\,\dot{M}_{\rm Edd}$. However, even under this high accretion limit,
$\gtrsim6\,M_\odot$ BHs are typically challenging to significantly spin up and
generate detectable associated EM signals. Our population simulations suggest
that different accretion limits have a slight impact on the ratio of tidal
disruption events. However, as the accretion limit increases, the EM
counterparts from the cosmological BHNS population can become bright overall.
