If hypothetically, the vantage point would move 2AU along an orbit around the star being measured, would that make it seem as if the other star moved twice the distance over the back-drop?
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The other star would have a significant proper motion.
Over the course of one orbit, it would appear to travel right around the sky once. The variation in position due to the Earth's orbit of the sun would be superimposed on top of this motion.
Except that the proper motion of the star, and its parallax would be much greater than for any real star, this is not really different to the way in which the parallax motion of a star is superimposed on its proper motion.