  Cavity optomechanics is showing promise for studying quantum mechanics in
large systems. However, smallness of the radiation-pressure coupling is a
serious hindrance. Here we show how the charge tuning of the Josephson
inductance in a single-Cooper-pair transistor (SCPT) can be exploited to
arrange a strong radiation pressure -type coupling $g_0$ between mechanical and
microwave resonators. In a certain limit of parameters, such a coupling can
also be seen as a qubit-mediated coupling of two resonators. We show that this
scheme allows reaching extremely high $g_0$. Contrary to the recent proposals
for exploiting the non-linearity of a large radiation pressure coupling, the
main non-linearity in this setup originates from a cross-Kerr type of coupling
between the resonators, where the cavity refractive index depends on the phonon
number. The presence of this coupling will allow accessing the individual
phonon numbers via the measurement of the cavity.
