First observation that the Sun and Jupiter (and friends) move around a common barycenter? Answers to the question How did Kepler determine the orbital period of Mars? describing careful observations centuries ago got me thinking.

What was the first analysis of observations that directly demonstrated that the Sun and the outer giant planets were moving around a common barycenter rather than all planets rotating around a fixed Sun?

I am not sure if First observation that the movement of a planet or asteroid in its orbit was affected by another planet? is a completely separate question, or if this confirmation happened all at once, but I've currently asked it separately.
This idea would be a direct reduction from Newton's laws of motion.  In his Principia he states "Hence the common centre of gravity of the earth, the sun, and all the planets, is to be esteemed the centre of the world".  So that would be the first evidence of the Sun moving outside of a geocentric universe.  Of course he wouldn't have known about Uranus and Neptune at the time.