Cavity optomechanics is showing promise for studying quantum mechanics in large systems. However, the 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 can be exploited to arrange a strong radiation-pressure-type coupling g₀ 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 g₀. Contrary to the recent proposals for exploiting the nonlinearity of a large radiation-pressure coupling, the main nonlinearity 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.