There are apparently some extensions of the Standard Model that allow for Lorentz symmetry to be violated, although from what I understand the symmetry is broken by spontaneous symmetry breaking which means the theory would have been symmetric in the era when the forces were unified, and the symmetry was broken by a random decay to different vacuum state. So in this case there wouldn't be any asymmetry in the fundamental laws of physics, just in the particular vacuum state which our region of the universe has, which was fixed by contingent events in the past. The article "Breaking Lorentz Symmetry" from PhysicsWeb discusses such Lorentz-symmetry-violating extensions of the Standard Model in more detail, here's the section explaining the relevance of spontaneous symmetry breaking:
The alignment of a magnet is a classic example of what is called
spontaneous symmetry breaking. The interactions of the individual
dipoles in a magnet do not depend on any particular direction, and
their dynamics are rotationally invariant. For the magnet to form,
however, the dipoles must spontaneously align in some direction, which
"spontaneously breaks" the rotational symmetry.
In 1989 Kostelecky and Stuart Samuel of the City University of New
York showed that string theory allows for Lorentz symmetry to be
spontaneously broken in the early universe. If Lorentz symmetry is
spontaneously broken, small relic background fields - which are called
tensor-valued vacuum expectation values - would permeate the universe
and point in spontaneously chosen directions. An elementary particle
in the presence of one of these relic fields would then experience
interactions that have a preferred direction in space time. In
particular, there could be preferred directions in 3D space in any
fixed reference frame, such as an Earth-based laboratory.
At a fundamental level, Lorentz symmetry would still hold dynamically,
and all interactions would remain invariant under observer Lorentz
transformations. However, the presence of the relic fields would break
the particle Lorentz invariance, leading to variations in physical
interactions as the motion or orientation of a particle changes with
respect to the relic fields.
Not sure if there'd be reason to expect the preferred frame of such a relic field to be the same as the CMB frame, though.
By the way, the difference between "observer" Lorentz symmetry and "particle" Lorentz symmetry discussed in that article, with "observer" symmetry being about the fundamental laws but "particle" symmetry being possibly broken by relic fields, is also discussed on pages 275-276 of the book Out of this World: Colliding Universes, Branes, Strings, and Other Wild by Stephen Webb--those pages can be read on google books here.
That book specifically says that it is in the context of M theory, the speculated unification of various string theories, that it's expected that observer Lorentz symmetry would be preserved but particle Lorentz symmetry might be violated. I also found another reference to the assumption that the fundamental laws of M theory are expected (though only as an ansatz or educated guess) to be invariant under the Lorentz transformation, on p. 368 of the book *Duality and Supersymmetric Theories which refers on p. 368 (viewable here) to an argument that "M-Theory ... is a fully Lorentz invariant theory in 11 dimensions." But in other approaches to quantum gravity, I think physicists have less confidence that the theory will incorporate some form of Lorentz invariance. For example, see p. 8 of this paper which discusses possibilities in an approach known as emergent gravity:
While the previous discussion refers to the standard framework (GR
plus QFT), different outcomes for the speed of light postulate can be
envisaged when departures from GR are taken into account. The search
for such departures has been boosted in recent years by rising of the
emergent gravity paradigm. Within this framework it is in fact very
natural to see also Lorentz invariance as an emergent spacetime
symmetry broken at high energy. Indeed, we have nowadays several toy
models where a finite speed of propagation can emerge in systems
having no fundamental speed limit [34]. For example, this is the case
with the speed of sound in Newtonian (non-relativistic) condensed
matter systems. Individual particles of the system can move at
arbitrarily large speeds; however, collective density disturbances of
wavelengths larger than the inter-particle distance, all propagate at
the same finite speed, the speed of sound.
If electromagnetic fields were emergent collective excitations of an
underlying system with the speed of light playing the role of the
speed of sound, then, any particle or excitation moving at speeds
large than c would slow down by emitting electromagnetic radiation,
much as in the Cˇerenkov effect. The speed of light will appear as
insurmountable in practice. This perspective offers an answer to the
question with which we started this essay: Because all the physics
that we know of, even that in accelerators, is low-energy physics and
all the known fields collective variables of a yet unknown underlying
system. Maybe it is allowed to travel faster that light, but only for
high-energy beings.
The breakdown of Lorentz invariance generally manifest itself via
dispersion relations for matter modified at energies close to the
Planck scale, about $10^{19}$ GeV.
And this recent paper says:
the fate of relativistic symmetries in the Planckian regime has attracted interest from other angles (see e.g. 6, 7 and 8). Relativistic symmetries may be left unscathed by the new structures at the Planck scale (e.g. 9), but there are at least two other possibilities. Planck-scale effects may break relativistic invariance, introducing a preferred-frame [10], [11], [12], [13] and [14]; or they may deform the relativistic symmetry transformations, preserving the relativity of inertial frames [15], [16], [17], [18], [19] and [20].
Looking at the references that follow the phrase "introducing a preferred-frame" in the paper, it looks as though they all deal with approaches to quantum gravity distinct from M theory, such as loop quantum gravity.
