Let's clear up an apparent misunderstanding here. A remote planet's effect on Earth's precession is proportional to the gravitational torque it imposes on our equatorial bulge. That torque is not necessarily proportional to the outer planet's displacement of the Sun and inner planets relative to the barycenter.

Let's analyze a simplified example starting with only the Sun, Earth and Jupiter. The presence of Jupiter displaces the Sun about half a million miles from the barycenter. Earth's orbit remains roughly centered on the Sun but is perturbed by Jupiter's gravity, which also imposes a precession-inducing torque on our planet.

Now let's add another planet of Jupiter's mass ten times farther out. To maintain dynamic balance, the Sun will now be about 10 times farther from the barycenter, with Earth's orbit going along with the Sun. However, the new planet's contribution to the torque on Earth will be only about 1/1000 that of Jupiter, because that effect drops off as the cube of the distance, for the same reason as the dropoff of the tidal force. This planet's perturbation of Earth's orbit is reduced by a comparable amount.

Conclusion: A distant planet's perturbation of an inner planet in any way is disproportionately small in proportion to its contribution to the barycenter position.

My rough estimates are that Saturn's torque is only a few per cent of Jupiter's. Uranus contributes about 1/1000 as much as Jupiter, and Neptune even less. Venus averages about as much as Saturn, and at inferior conjunction peaks at many times that of Jupiter.

More on this later. I have an appointment coming up and I must quit for now.