  Self-interacting dark matter (SIDM) cosmologies admit an enormous diversity
of dark matter (DM) halo density profiles, from low-density cores to
high-density core-collapsed cusps. The possibility of the growth of high
central density in low-mass halos, accelerated if halos are subhalos of larger
systems, has intriguing consequences for small-halo searches with substructure
lensing. However, following the evolution of $\lesssim 10^8 M_\odot$ subhalos
in lens-mass systems ($\sim 10^{13}M_\odot$) is computationally expensive with
traditional N-body simulations. In this work, we develop a new hybrid
semi-analytical + N-body method to study the evolution of SIDM subhalos with
high fidelity, from core formation to core-collapse, in staged simulations. Our
method works best for small subhalos ($\lesssim 1/1000$ host mass), for which
the error caused by dynamical friction is minimal. We are able to capture the
evaporation of subhalo particles by interactions with host halo particles, an
effect that has not yet been fully explored in the context of subhalo
core-collapse. We find three main processes drive subhalo evolution: subhalo
internal heat outflow, host-subhalo evaporation, and tidal effects. The subhalo
central density grows only when the heat outflow outweighs the energy gain from
evaporation and tidal heating. Thus, evaporation delays or even disrupts
subhalo core-collapse. We map out the parameter space for subhalos to
core-collapse, finding that it is nearly impossible to drive core-collapse in
subhalos in SIDM models with constant cross sections. Any discovery of
ultra-compact dark substructures with future substructure lensing observations
favors additional degrees of freedom, such as velocity-dependence, in the cross
section.
