ABSTRACT Super-Eddington accretion of neutron stars (NSs) has been suggested both observationally and theoretically. In this paper, we propose that NSs in close-orbit binary systems with companions of helium (He) stars, most of which systems form after the common-envelope phase, could experience super-Eddington stable Case BB/BC mass transfer (MT), and can sometimes undergo accretion-induced collapse (AIC), resulting in the formation of lower mass-gap black holes (mgBHs). Our detailed binary evolution simulations reveal that AIC events tend to happen if the primary NSs have an initial mass ${\gtrsim} 1.7\, {\rm M}_⊙$ with a critical accretion rate of ≳300 times the Eddington limit. These mgBHs would have a mass nearly equal to or slightly higher than the NS maximum mass. The remnant mgBH–NS binaries after the core collapses of He stars are potential progenitors of gravitational-wave (GW) sources. Multimessenger observations between GW and kilonova signals from a population of high-mass binary NS and mgBH–NS mergers formed through super-Eddington stable MT are helpful in constraining the maximum mass and equation of state of NSs.