As @Ben Crowell commented relativistic mass is not used in general relativity (GR), and is also not a preferred term in special relativity (SR)
In SR it came from the equation for energy to include the kinetic energy, as E=$mc^2$, so m was identified as $\gamma m_0$, with $\gamma$ the Lorentz factor. In GR it does not come out so simply, and the stress energy tensor is the source of gravity. The diagonal of the stress energy tensor are E and $p^i$, the momentum, and there are other stress terms.
If you had two bodies (galaxies or whatever, if you consider them point particles), they each will have an effect on the other, and if you want to write the equations under their GR gravity, they become two sets of nonlinear differential equations. This so called two body problem (unless one mass is much larger than the other) cannot be solved exactly in GR. The equivalent two body problem in Newtonian mechanics has an exact solution, but not in GR. It was solved approximately using a parametrized post-Newtoninan (PPN) approximation, and more exactly using pretty complex numerical methods (for the merging two black hole two problem it took 40 or so years to get it right).
The simplistic view is yes they attract each other, and yes their momentum in each case acts as a source of gravity, but the results are not simple. For the cosmological case I have not seen a solution for two galaxies, but it must be there. You'd take the cosmological metric, which is expanding, and treat their two body GR gravity as an additional perturbation (assuming it is much weaker than the cosmological effect, which for superluminal expansion would be the case unless you had some real monster galaxies), and of coUrse the cosmological expansion between the two will be slowed very little by their two body gravity. On the other hand if they are not that far from each other, the cosmological effect will be much smaller, they will not be superluminal, and they will tend to orbit each other.
In cosmology the effect of the expansion is overwhelmed by the effect of nearby masses up to maybe 100 Mpsecs. Once you start detecting the redshift of the expansion methodically, local effects from nearby galaxies have less effect. In fact, we know there are some cosmological regions with slightly more mass density than others, as the homogeneity is not exact or 100%. Analyzing inhomogeneities/anisotropies in the cosmic microwave background (which are very small) is a continuing research activity, and gives us information that is also used to study and model star and galaxy formation. See for instance it's relationship to astronomy in