Implications of the axis of evil in Big Bang theory and cosmological inflation? According the the axis of evil feature of the CMB map verified twice by space missions our home location is on this center axis of the CMB map! Making our home location sort of the center of the observable Universe!
This is regarded today as one of the unsolved cosmological problems:

Axis of evil: Some large features of the microwave sky at distances of over 13 billion light years appear to be aligned with both the motion and orientation of the solar system. Is this due to systematic errors in processing, contamination of results by local effects, or an unexplained violation of the Copernican principle?
In another study, in June 2020, the 'axis of evil' observation was repeated in a study conducted by Lior Shamir, which verified the same result measuring galaxy rotations.[22][23][24] Shamir’s paper suggests that the observation could be explained by the hypothesis that the early universe was less random than it is today, and that it was spinning on its axis. Shamir commented: "We have two different sky surveys showing the exact same patterns, even when the galaxies are completely different. There is no error that can lead to that. This is the universe that we live in. This is our home."

However, nobody seems to address (besides the Copernican Principle violation) the "elephant in the room" here which is:
Does this unexplained phenomenon have direct implications to the the Big Bang theory and Cosmological inflation and if yes which are these?
 A: 
Does this unexplained phenomenon have direct implications to the the
Big Bang theory

Not necessarily, although it depends to some extent on how you define the contours of Big Bang theory.
It does not (at least as a matter of necessity) contradict the notion that all of the mass-energy of the universe was confined to a very small space about 13.7 billion years ago that rapidly expanded, that the universe continues to expand, and that the decreased density and temperature of the universe caused it to evolve in particular ways (like Big Bang Nucleosynthesis) that gave rise to our current universe. This does not inherently require spherical symmetry, homogeneity, or a lack of anisotropy.
Explosions that are disk-like rather than spherical are common place in the ordinary world of classical physics and there is nothing inherent in the Big Bang Theory that precludes a non-spherically symmetric "bang" in that context.

and Cosmological inflation and if yes which are these?

The "axis of evil" if it is more than a systemic or methodological artifact of how we can gather astronomy data from our current location in the universe is evidence that the universe is anisotropic and not homogeneous.
One of the main purposes of cosmological inflation as an explanatory theory is to explain why the universe is isotropic and homogeneous to a greater extent than one might expect in the absence of cosmological inflation:

Many physicists also believe that inflation explains why the universe
appears to be the same in all directions (isotropic), why the cosmic
microwave background radiation is distributed evenly, why the universe
is flat, and why no magnetic monopoles have been observed.

While it doesn't necessarily rule out any and all possible cosmological inflation theories, almost all of them assume (in part for ease of analysis but in part for other sound reasons) spherical symmetry as a core raison d'etre.
But, if the axis of evil is a real phenomena, then it almost necessarily follows that this structure was seeded prior to or during cosmological inflation. This is because cosmological inflation is used to explain the large scale structure of the universe and the "axis of evil" if real is the largest scale observed feature of the large scale structure of the universe. So, if the axis of evil is real this implies that a very different mechanism, perhaps a variant of cosmological inflation or perhaps something entirely different, explains the qualitative features of the universe we observe which cosmologists frequently assume are due to one of the hundreds of competing cosmological inflation theories that has been proposed so far.
Of course, it is also worth noting that while the Big Bang theory in some permutation or another has almost universal acceptance among astrophysicists, cosmological inflation does not have consensus support, let alone consensus support around any particular mechanism of that theory. Even one of the original inventor/discoverers of the concept, Alan Guth, now seriously doubts it.
An alternative explanation is that cosmological inflation which in the most common versions of this class of theories cannot explain this observation, is wrong and that the features of the large scale structure of the universe which cosmological inflation seeks to describe have a different source.
It isn't fruitful to speculate too deeply on what alternatives to cosmological inflation could involve, even though there are a handful of proposals out there, such as the torsion-based cosmological model proposed by Nikodem Poplawski mentioned by @Edouard, since none of them have wide acceptance or have been seriously vetted by others in the field. If the "axis of evil" observations start to look like they are real and not just methodological artifacts, then in due time new theories will develop to explain it whose features are hard to predict if one is not omnipotent.

nobody seems to address (besides the Copernican Principle violation)
the "elephant in the room"

These seem like two sides of the same coin to me, rather than distinct concerns. Cosmological inflation and the Copernican Principle are deeply intertwined. The former is a common and standard explanation for the latter (although not the only one).
A: It appears that the results found by Lior Shamir are supportive of the torsion-based cosmological model proposed by Nikodem Poplawski (whose numerous 2010-2021 preprints can be found by his name on Cornell University's Arxiv website) in 2010.  His 2010 paper, "Cosmology with torsion", is subtitled "An alternative to cosmic inflation", but has been interpreted as a version of inflation, even though it lacks the "inflaton" particles that remain entirely  hypothetical and characterize older inflationary models.
In Poplawski's model, any large rotating star (-all stars are usually considered to have at least some faint residual rotation) that has expended its nuclear fuel, and consequently lacks the radiation pressure that had prevented its collapse, collapses to form a black hole.  During collapses involving a certain relation between the Compton radius and the Compton time, new particles materialize from the star's gravitational field, and their contact with the vastly larger stellar fermions spins them outward, forming a new "local universe" whose shape Poplawski has analogized to "the skin of a basketball".
Spherical universes are usually considered to have a finite duration, although it may (depending on other aspects of their model) be long enough to favor the repetitive conditions whose discovery may be the root and purpose of science itself.  Poplawski describes the typical local universe of his model as "expanding indefinitely", which leaves wiggle room between the sort of place where the survivor of a sphere's rotation in spacetime might murder their own grandpa and the sort of place (like ours) where they wouldn't.
Far from being the very special place that the geocentricity uprooted by Copernicus would've otherwise inflicted upon us, Poplawski's model would reveal our locality to be as common as dirt, or, at least, as common as particles of it.  Why?  Because each local universe would eventually contain its own rotating stars, that its own inhabitants might consider to be really really large, and some of them would repeat the process, presumably on sequentially-decreasing scales.
A couple of problems stand in the way of his model's acceptance:  One of them, because that model is past- and future-eternal, is philosophical or religious creationism, and another is mathematical complexity, as his model utilizes 1929's Einstein-Cartan Theory (as subsequently modified by Sciama and Kibble), which requires fermions (matter particles) to have some tiny spatial extent, a few orders of magnitude greater than the Planck length. (In utilization of 1915's General Relativity--an earlier result of Einstein's extraordinary perceptions--fermions are considered to be "point-like".)
