Good question re dynamics. Yes, I talked about Muller’s work in my book LS, and looked at a few different scenarios. The common approach to determining the distance of a companion in relation to the solar system is to constrain the sun’s motion to zero and then plot based on Kepler’s laws, which implies everything pretty much moves at the same speed and the longer the periodicity the more distant the object and vice versa. So you are right, suggesting a periodicity that produces the observable known as precession (a cycle of ~24,000 to 26,000 years), requires the object be very close, under this static Sun scenario. Those calculations are detailed on our website.
But we have also done the calculations based on the assumption the solar system moves. We observe the sun race through space against the background stars at about 50”p/y, yet every since Copernicus this observable has been assumed to be false, the artifact of a wobbling earth. Remember Copernicus had just dethroned the earth from being the center of the Universe (to the consternation of the Church) and put the immovable Sun in its place, so he could not say that the Sun moved! He assumed the earth wobbled, and ever since people have been trying to come up with a precession equation that works. Noticing all the problems associated with precession explanations, and finding no solutions, we have taken the approach the sun really does move.
This approach should not be all that novel but since luni-solar precession theory constrains SS motion to zero, you will not find any work on this in the literature. Bottom line, there are several viable scenarios that suggest the sun could be gravitationally bound to a nearby visible star, it just means that our solar system is moving faster than it is currently allowed under luni-solar constraints. Even Sirius at ~8.6ly, with a center of mass~6.6ly is viable if the SS is moving at 430km/s. This seems high under the current paradigm but we do know that many stars move at incredible speeds (some twice this number) and Reg Cahill, an astrophysicist our of Australia has independently calculated 430km/s the speed of our SS versus the CMB (but not sure about this methodology). And to your point, it does NOT require high eccentricity (highly elliptical orbit).
Personally, I like the Sirius scenario for many reasons, especially because it would seem to explain the incline of Pluto (whose orbit period is also at a perfect 5:1 ratio of the Sirius B orbit periodicity, which may have a pendulum effect) and the highly elliptical (and weird) orbit of Sedna, which happens to be in 2:1 resonance with the precession periodicity (companion orbit). These resonances are expected and necessary in large dynamic systems, especially for the outer dwarfs. Mike Brown at Caltech (the man who killed Pluto) is apparently still looking for a way to explain Sedna's orbit.
Chinese Philosopher – Lao Tzu