This issue of kinematics has been well studied. To make a long story short, you are missing a reference frame. Just as the moon's orbit is attached to the earth as the earth orbits the sun, yet you won't see a greatly different sky from the moon than you will the earth, so too is the solar system moving along with its binary system (and probably even the local stellar neighborhood as it all goes around the GC, which is another issue), and you should not expect to see anything too different than you now see (partially because of the great distances of the stars) other than the sun moving through the constellations at about 50"p/y (which some call precession : )). Anyway, we have detailed discussions on this on the BRI website, which I think you will find useful. While you are there I would appreciate if you would check our math on the latest paper "Predicting Changes in Earth Orientation - Dynamic Versus Static Solar System Model", as presented at the AGU, and please let me know if you find any mistakes. The math says the moving solar system model is far more accurate than the luni-solar model in predicting changes in EO - the numbers are unbiased and they work!