There's only one barycenter for the solar system, which includes the impact from every planet, asteroid, comet, and space probe in the solar system. The path the sun takes around that barycenter is mostly a circle because the Sun-Jupiter interaction completely dominates the physics of the system, but it's a bit wobbly due to all the interacting forces of the other bodies (mostly Saturn).
You can calculate the location of the barycenter between the sun and any other body in the system, but the sun doesn't continually switch which one of those it's paying attention to. It has to obey all of them all the time, leading to a complicated wobbly path around the single Solar System barycenter.
But it doesn't stop there, because all those other bodies are pulling on each other as well -- even as Jupiter is pulling on the sun and Saturn is pulling on the sun, those two are pulling on each other, making their orbits wobble as well, which feeds back into the sun's wobble. It's an almost infinitely complex interaction of forces.