I do not understand the quoted part... After a quick look it seams that in the Cooperstock paper it is taken as an assumption that the scale factor may vary within the solar system. The computations are then based on this assumption:
However, it is reasonable to pose the question as to whether there is a cut–off at which systems below this scale do not partake of the expansion. It would appear that one would be hard put to justify a particular scale for the onset of expansion. Thus, in this debate, we are in agreement with Anderson (1995) that it is most reasonable to assume that the expansion does indeed proceed at all scales.

I was not able to find this Anderson paper, however. My understanding is that without an homogeneous and isotropic dark energy permeating the whole space at all scales above the Planck lenght, there should be a cutoff for the expansion of space. This should be determined as a function of some characteristic length at which matter distribution starts to be homogeneous and isotropic.
At that lenght there should be then a coupling between the cosmological solution and the Schwarzschild or axiallysymmetric solution at smaller scales. I would guess that the galactic distribution of matter (and therefore an axiallysymmetric solution), rather than the solar system, is the solution that has to be coupled to the cosmological one. I see no reason a priori to think that this coupling would imply an influence of the external metric on the internal one at arbitrary short distances.
Things are different, however, if there exists a homogeneous and isotropic distribution of dark energy at arbitrary small scales above the Planck length. I can imagine that this could imply an expansion of space at every scale, that may be however not noticeable within the solar system.
This is my personal view of this, that seams not to be the "standard" one. At least not the one of Anderson, Cooperstock and Wright. But I fail to see the arguments that may invalidate it.