Jupiter isn’t pulling the sun out of stability. There isn’t really any notion of the sun’s stability of position anyway. Not within the solar system. There are instabilities in the orbit of stars around the galaxy, and very occasionally stars can be ejected from the galaxy. Something that takes of the order of billions of years to happen.
Another way of looking it is that the barycentre is just the centre of mass of the two bodies. That moves linearly with the mass ratio of the bodies.
Gravitational attraction however drops with the square of the distance. The Earth is so much closer to the sun than Jupiter that even when Jupiter is nearest the effect of its gravity on the Earth is attenuated hugely.
Sun → Earth = 1.5 \times 10^{11}m
Sun → Jupiter = 7.50 \times 10^{11}m
Jupiter → Earth nearest = 6.0 \times 10^{11}m
distance squared ratio = 2.25/36 = 0.0625
Mass Jupiter =1.9 \times 10^{27} kg
Mass Sun =2 \times 10^{30}kg
So another factor of a thousand, to make Jupiter’s gravitational attraction to the Earth become, at best, 0.00006 times that of the Sun. It isn’t zero, and the effect is real, but it isn’t a concern.
There are quite a few videos out there that attempt to show how things really work. Here is one:
