Nice point about General Relativity! But you needn’t even take it that far, even under Newton’s laws of gravitation, it’s clear that the Earth doesn’t just orbit the Sun, but they in fact orbit each other. Therefore, strict heliocentrism or strict geocentrism are either equally wrong, or equally right, depending on your perspective :)
Actually, the differences are greater than that. Heliocentrism always ignores the rest of the universe while geocentrism always includes it. Your example above shows how the rest of the universe is ignored by focusing on the sun/earth CG.
Consider that the solar system is a mere point when compared to the universe. Then consider that it does not take much gravitational imbalance in the universe to move the CG from somewhere inside the sun (for heliocentrism) to inside the earth (for geocentrism).
This is what Ernst Mach understood under classical physics and what Einstein, Hoyle, Born and Ellis understood under GR.