  Self-interacting dark matter (SIDM) has been proposed to solve small-scale
problems in $\Lambda$CDM cosmology. In previous work, constraints on the
self-interaction cross-section of dark matter have been derived assuming that
the self-interaction cross-section is independent of velocity. However, a
velocity-dependent cross-section is more natural in most theories of SIDM.
Using idealized $N$-body simulations without baryons, we study merging clusters
with velocity-dependent SIDM. In addition to the usual rare scattering in the
isotropic limit, we also simulate these systems with anisotropic, small-angle
(frequent) scatterings. We find that the collision-less brightest cluster
galaxy (BCG) has an offset from the DM peak that grows at later stages.
Finally, we also extend the existing upper bounds on the velocity-independent,
isotropic self-interaction cross-section to the parameter space of rare and
frequent velocity-dependent self-interactions by studying the central densities
of dark matter-only isolated haloes. For these upper-bound parameters, the
DM-BCG offsets just after the first pericentre in the dark matter-only
simulations are found to be $\leq$ 10 kpc. On the other hand, because of BCG
oscillations, we speculate that the distribution of BCG offsets in a relaxed
cluster is a statistically viable probe. Therefore, this motivates further
studies of BCG off-centring in hydrodynamic cosmological simulations.
