The Sol-Jupiter system would have a bary center just 7% outside the surface of Sol. The effect of all the Gas Giants together can either center the syster in Sol's core or move the barycenter 120% outside Sol.

The really weird thing is that the part of stars outside the core is more like an atmosphere. If the star gets hotter, the parts outside the core can expand. This is happening slowly as Sol's core fills up with Helium and becomes denser, which fuses Hydrogen faster. So despite weighing less, the Sol-Jupiter barycenter will be engulfed within Sol's envelope. Once Sol stops fusing Hydrogen in it's core, the core will shrink and heat up, fusing Hydrogen in a shell around the core, which will cause the envelope to grow and engulf Venus and possibly Earth directly, and definitely contain the full system's barycenter. After that it will release a bunch of mass in a planetary nebula, which will cause it to shrink a lot, and the remaining planets will probably orbit much farther out, which would throw the barycenter waaaayyyyyy outside of the white dwarf left over.