  Theoretical models for the expected merger rates of intermediate-mass black
holes (IMBHs) are vital for planned gravitational-wave detection experiments
such as the Laser Interferometer Space Antenna (LISA). Using collisionless
$N$-body simulations of dwarf galaxy (DG) mergers, we examine how the orbital
decay of IMBHs and the efficiency of IMBH binary formation depend on the
central dark matter (DM) density profile of the merging DGs. Specifically, we
explore various asymptotic inner slopes $\gamma$ of the DG's DM density
distribution, ranging from steep cusps ($\gamma=1$) to shallower density
profiles ($\gamma<1$), motivated by well-known baryonic-feedback effects as
well as by DM models that differ from cold DM at the scales of DGs. We find
that the inner DM slope is crucial for the formation (or lack thereof) of an
IMBH binary; only mergers between DGs with cuspy DM profiles ($\gamma=1$) are
favourable to forming a hard IMBH binary, whereas when $\gamma<1$ the IMBHs
stall at a separation of 50-100 pc. Consequently, the rate of LISA signals from
IMBH coalescence will be determined by the fraction of DGs with a cuspy DM
profile. Conversely, the LISA event rates at IMBH mass scales offer in
principle a novel way to place constraints on the inner structure of DM halos
in DGs and address the core-cusp controversy. We also show that, with spatial
resolutions of $\sim$0.1 kpc, as often adopted in cosmological simulations, all
IMBHs stall, independent of $\gamma$. This suggests caution in employing
cosmological simulations of galaxy formation to study BH dynamics in DGs.
