I am familiar with frame dragging, but the best tests I am aware of show no measurable change in GPS clocks in the direction(orbit) of our solar system with respect to the center of our galaxy (approx. 250 km/s)
As the Earth is but a small chunk of rock compared to the solar system, and it compared to the surrounding stars and dust in our local arm of the galaxy, I do not think the motion of the Earth would register above the motion of the whole arm of our galaxy.
So my question is, would it register if A were the only mass and if it were the only mass would the registration indicate motion?