The GR doubling effect on horizontal radiation means that the rate of "free fall" is not the same for different parts of the Sun. Where there is a greater proportion of the energy or mass content that is radiation the rate of fall is faster than where there is a smaller proportion of radiation. The centre has a greater proportion of radiation than the surface.

This is somewhat like tidal effects where the rate of fall varies over the extent of a body, except that tidal effects create a stretching in one direction and a squeezing at all directions at right angles to that. Whereas this effect creates a displacement of the interior relative tothe surface, leading (in combination with rotation) to a slight convection current with a toriodal shape.

Additionally, tidal effects are dependent on M/R^3 whereas this effect is dependent on M/R^2. So you are right that tidal effects are tiny for the galaxy. However this effect also depends on T^2 and for the galaxy (where we remain N and S of the plane for 30 million years at a time) this becomes quite huge.