Hello Guruk-hai - and welcome to the forum.
I like the way you formulated your original question which basically asks: “what can be ruled out? Can we rule out that precession may be explained by something other than the Earth’s motion around the PVP orbit?”
You see, I’m very fond of this good’ol Sherlock Holmes quote:
“When you have eliminated the impossible, whatever remains, however improbable, must be the truth”
In fact, I have been abiding scrupulously to this ‘golden rule’ throughout my TYCHOS investigations - and the question of the observed stellar motions (be they daily, yearly or secular).
So let’s see: could the entire fixed star sphere be moving - instead of the Earth? Let’s start by probing the observed, daily motion of the stars. Of course, classic ‘old-school geocentrists’ believe that all the stars revolve around in unison around the Earth every 24 hours, Can we eliminate this proposition beyond reasonable doubt? Yes we can. Today we know that every single star has its own so-called ‘proper motion’ (in all kinds of “x,y,z” directions), meaning that we may rule out the idea that they are part of a single ‘block’ - or ‘crystal sphere’, as first proposed by Anaximenes in the 6th century BC :
“The most enduring feature of Anaximenes’ cosmos was its conception of the stars being fixed on a crystal sphere as in a rigid frame, which became a fundamental principle of cosmology down to Copernicus and Kepler.”
So today we can firmly rule out the rather bizarre idea that all the stars would revolve in unison around us every day - due to the observable proper motions of the stars, in all sorts of directions. As it is, for the very same reason, we can also rule out the Copernican / heliocentric idea that our (nearmost) stars are all revolving ‘in unison’ with us around the centre of the galaxy. To be sure, this is basically what Copernicans are telling us: “all our surrounding stars seem to be almost stationary, since they all move at about 800000km/h (!) around the galaxy - along with our Solar System”. This idea is of course just as bizarre as the geocentric theory! For why would the stars be seen to move in all sorts of different “x,y,z” directions while ALSO moving in unison around the galaxy with our system?
But let me return to your specific question, namely: “could the stars possibly move around us in 25344 years - rather than Earth itself moving around its PVP orbit, as posited by the TYCHOS model?”
Well, as expounded and illustrated in Chapter 26 of my book, Cornelius Kapteyn concluded at the end of his illustrious career that there must be two ‘streams of stars’ moving in opposed directions. Kapteyn was the top world expert of his time in statistical analysis of stellar motions and his conclusions caused a huge stir within the astronomy community (which has long been ‘forgotten’ and swept under the rug).
The TYCHOS model provides the simplest (and I dare say, the ONLY plausible and logical) explanation for Kapteyn’s findings: since the Earth slowly moves along its PVP orbit, observers ‘on either side’ of the Earth will in fact see (during, say, a 100-year period) our surrounding stars moving in ‘opposed directions’ - and naturally so. This is illustrated in my below graphic;
Note: both Joe and Jim are depicted as standing with their backs facing the viewer and looking at stars located above their respective southern horizons.
In other words, Kapteyn was absolutely right - yet he never could make any sense of his findings, since he was unaware of the Earth’s motion along its PVP orbit.
Mind you, in my new TYCHOS book you will find several other demonstrations in answer to your question (i.e. that is its Earth’s motion - rather than the stars’ - that is responsible for the Great Year). However, I will stop here for now - hoping you will set aside some time to read the book in its entirety, if you haven’t done so already. 
