  Galactic nuclei are potential hosts for intermediate-mass black holes
(IMBHs), whose gravitational field can affect the motion of stars and compact
objects. The absence of observable perturbations in our own Galactic Centre has
resulted in a few constraints on the mass and orbit of a putative IMBH. Here,
we show that the Laser Interferometer Space Antenna (LISA) can further
constrain these parameters if the IMBH forms a binary with a compact remnant (a
white dwarf, a neutron star, or a stellar-mass black hole), as the
gravitational-wave signal from the binary will exhibit Doppler-shift variations
as it orbits around Sgr A$^\star$. We argue that this method is the most
effective for IMBHs with masses $10^3\,M_\odot\lesssim M_{\rm IMBH}\lesssim
10^5\,M_\odot$ and distances of $0.1$ mpc to $2$ mpc with respect to the
supermassive black hole, a region of the parameter space partially
unconstrained by other methods. We show that in this region the Doppler shift
is most likely measurable whenever the binary is detected in the LISA band, and
it can help constrain the mass and orbit of a putative IMBH in the centre of
our Galaxy. We also discuss possible ways for an IMBH to form a binary in the
Galactic Centre, showing that gravitational-wave captures of stellar-mass black
holes and neutron stars are the most efficient channel.
