You seem to be groping towards the fact that the gravitational three-body problem is, in general, not solvable.
We can get away with saying "the sun is at one focus of an elliptical orbit" in the solar system because the sun is so much larger than anything else around. The sun is 1000 times more massive than Jupiter, so the sun-Jupiter barycenter is about 1/1000th of the way from the sun's center of mass to Jupiter's center of mass. That's actually closer to the sun's surface than to its center: Jupiter's orbital radius of 5 AU is about 1000 solar radii.
Generally, as the mass hierarchy becomes more equitable, Kepler's approximation of "sun at one focus" becomes poorer, and the long-term behavior of the system becomes more chaotic. We concentrate on the perturbation approach to $n$-body systems because it's possible to get answers, and those answers happen to be useful for our particular solar system.
If you wanted to have a look at whether Earth's orbit is better described as an ellipse about the sun's center of mass or as an ellipse about the sun-Jupiter barycenter, you could try and dig up a table of perihelion distances over the past few decades. Because Earth's orbit is eccentric, Earth is about seven solar radii nearer to the sun in January than in July.
In years when Jupiter is at or near opposition in January (i.e. in Gemini or Taurus, as in 2013–2014) then the Sun will be on the July side of the sun-Jupiter barycenter: a little more distant at perihelion and a little nearer at aphelion. Perhaps the difference between Earth's perihelion and aphelion distances will only be about six solar radii in those years.
By contrast, when Jupiter is near opposition in July (as, I suppose in 2008-ish and 2020-ish) then the sun will swing to the January side of the barycenter and the contrast between Earth's perihelion and aphelion distances will be more pronounced: maybe eight solar radii difference.
On the third hand, if it's a better approximation to say (as I think you do) that the Earth's orbit is about the sun's center of mass and Jupiter be damned, then you should see comparable differences between perihelion and aphelion distance regardless of where Jupiter is.
On the fourth hand, it may be the least unreasonable thing to say is that the sun-Earth barycenter, esssentially indistinguishable from the center of the sun, orbits the sun-Jupiter barycenter … in which case the differences I was thinking of would become quite challenging to measure.
This is a small effect you're asking to suss out: you want to take the differences between two pairs of large numbers (sun-Earth distances at different times of the year) and compare those differences to a few percent. You would have to hunt down sun-earth distance data that you believe to four or five significant figures. You might be able to get this information out of free or commercial planetarium software, like Stellarium or Redshift, but you'll want to compare a few of them for consistency. Interesting question.
(I used Stellarium to look briefly at the difference between perihelion and aphelion differences for 2005—2015, odd years only because it was kind of labor-intensive, and saw year-on-year differences of order $0.01R_\text{sun}$, much less than my swinging-barycenter predictions above. I consider this interesting but not conclusive.)
