Well, my own understanding of Fermi-Walker transport in Minkowski space, can be found in my SRT FAQ:
https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf
It's of course clear that "rotation free spatial basis" is meant in a local sense, i.e., for an infinitesimal transport of an arbitrary vector, that is Minkowski perpendicular to the time-like tangent unit vector , it changes only by an rotation free Lorentz transformation, which is the idea of my derivation of the Fermi-Walker transport equation (1.8.6). Of course for a finite transport the spatial vector is necessarily rotating, if the time-like tangent vector changes direction, since you can see it as a composition of many rotation free Lorentz transformations with varying boost directions, and the composition of two rotation free Lorentz boosts in different boost directions is not rotation free anymore. This is the point of the Thomas precession (or mathematically spoken the Wigner rotation).
For a worldline of hyperbolic motion of a particle it's easy to see. The worldline starting for at is given by
The tangent vector along the curve is simply the four-velocity,
The dot stands for derivatives wrt. proper time, and I use the (+---) convention with for this posting (otherwise I get confused with the signs, although it's no problem to switch to the east-coast convention of course). The acceleration is
Now let's start with the basis , at . All we need to get the 2-bein along the hyperbolic world line is the Fermi-Walker transport of along this world line. Let's define . Then the FW-transport equation,
reads
The solution of course leads to
Indeed is not along anymore, although locally not rotating in the sense of the definition of Fermi-Walker transport.
Of course, in this case the spatial coordinate system (4-bein) along the hyperbolic world line is not rotating at all since all infinitesimal boosts are in one direction, here chosen as the -direction, and thus here we have no Wigner rotations.