It turns out that the Sun is only ever retrograde with respect to the solar system barycenter, if it's viewed from a relatively fixed point in space, such as another star. Viewed from any planet, the Sun and its barycenter will always be moving direct.The Sun orbits a barycenter also, caused by the orbiting Planets, especially Jupiter. I'm wondering if that causes a slight, periodic Solar retrograde movement
That's because the Sun's movement around the barycenter is slow, on about a 12 year cycle. Jupiter is the most massive planet, so Sun and Jupiter effectively orbit each other. Saturn is the next most massive planet, and when it aligns with Jupiter it tends to pull the Sun farther away from the barycenter, and when it opposes Jupiter it tends to pull the Sun towards it. The result is the Sun makes little loops in about a 20 year cycle (which is the period of Jupiter Saturn Conjunctions which happen every 20 years). Uranus is the third most massive planet, and its motion tends to distort the loops making them larger or smaller. The effect of all three planets on the Sun can be seen in the following diagram, showing the Sun's motion with respect to the barycenter over time:
In summary, because the motion is so slow and the Sun and barycenter are so close to each other, that's why both will always appear to be moving direct from any orbiting planet. Even outer bodies like Neptune and Sedna still move fast enough that it will make them always appear direct.
Astrolog supports computing the solar system barycenter as well as the physical Sun's body, and can display both at once for comparison. (And for Pluto, Astrolog can compute the position of Pluto's actual body, it's moon Charon's body, and compare both to the Pluto planetary system barycenter, the latter of which is what most programs are limited to computing.) For example, I can reproduce the solar system barycenter diagram above in Astrolog's orbit chart, which can track the orbital movements of objects: