  The abundance, distribution and inner structure of satellites of galaxy
clusters can be sensitive probes of the properties of dark matter. We run 30
cosmological zoom-in simulations with self-interacting dark matter (SIDM), with
a velocity-dependent cross-section, to study the properties of subhalos within
cluster-mass hosts. We find that the abundance of subhalos that survive in the
SIDM simulations are suppressed relative to their cold dark matter (CDM)
counterparts. Once the population of disrupted subhalos -- which may host
orphan galaxies -- are taken into account, satellite galaxy populations in CDM
and SIDM models can be reconciled. However, even in this case, the inner
structure of subhalos are significantly different in the two dark matter
models. We study the feasibility of using the weak lensing signal from the
subhalo density profiles to distinguish between the cold and self-interacting
dark matter while accounting for the potential contribution of orphan galaxies.
We find that the effects of self-interactions on the density profile of
subhalos can appear degenerate with subhalo disruption in CDM, when orphans are
accounted for. With current error bars from the Subaru Hyper Suprime-Cam
Strategic Program, we find that subhalos in the outskirts of clusters (where
disruption is less prevalent) can be used to constrain dark matter physics. In
the future, the Vera C. Rubin Observatory Legacy Survey of Space and Time will
give precise measurements of the weak lensing profile and can be used to
constrain $\sigma_T/m$ at the $\sim 1$ cm$^2$ g$^{-1}$ level at $v\sim 2000$ km
s$^{-1}$.
