Explaining rest electron magnetic field through Special Relativity

Physics is the discipline of studying nature and modelling it with mathematics, so that one can predict new observations and measurements.

In the process there are several frameworks where physics models exist, because of kinematic and other variables, and it can be shown mathematically that in the overlap region they merge or emerge from each other.

To have a mathematical model one has to impose extra axioms, called laws, principles,postulates in order to pick up those solutions that describe and predict data.

In the classical electrodynamics of Maxwell, it is not necessary for electric and magnetic fields to be related with motion in order to exist. There exist monopoles in the theory . The electric monopoles have been discovered in particle data, and it is the electron to start with (and quarks and muons ..) , but no magnetic monopole has been discovered (as yet?).

In classical theory it is not necessary to have a special relativity kinematics source for a magnetic field .

The world of particles is modeled with quantum mechanics, and within its axiomatic assumptions there is the table of particles with their charges and spins.

Axiomatic spin definiton for particles is necessary in order to have the law of angular momentum conservation at the particle level.

Your question

But I don't know how you would explain the magnetic field due to its internal magnetic dipole through Special Relativity because the charge is not moving at all.

Can you eliminate the electric field of the electron by special relativity motion?

The confusion comes by using classical electromagnetism where quantum modeling is needed. In the quantum frame , given a spin , nothing has to move, but it has been found that particle spin can be treated classically ( as there is continuity between classical and quantum) and since there is spin and mass, a magnetic moment a la classical EM can be calculated. Note that the classical calculation needs a correction factor.

It is useful to check the classical model behavior at the quantum level, it is not binding, the quantum is a different framework with different mathematics, and one has to be careful to keep to the correct framework when using mathematical tools from a larger one.

Electrons are photons are both just quanta of fields. The angular momentum of both can be understood as a bulk circular motion of the field. The electron field also has electric charge, and its magnetic field can be understood as arising from the same circular motion. This is covered by Hans C. Ohanian, "What is spin?", Am. J. Phys. 54 (6), June 1986 (online here). He credits F. J. Belinfante (1939) and W. Gordon (1928) for the original derivations.

The photon field was studied (as the electromagnetic field) before quantum mechanics was discovered, and was known to have angular momentum, and the angular momentum of photons is obviously just the quantized version of that classical property. Electrons were never understood as a classical field before quantum mechanics, and they were treated as point particles with "intrinsic spin" in the early days of QM, before quantum field theory came along. I suppose that's why you so commonly hear claims that electron spin is a mysterious property of point particles, while you rarely hear that about photons, even though in the modern picture they're closely analogous.

In Purcell's book, he gives an explanation of how the magnetic field arises due to wrong frames.

This approach seems to confuse many students. I do not recommend it for learning. It is nice to come back to once you already know the material.

If you look carefully at Purcell's actual derivation, he is not deriving the magnetic field, but rather the magnetic force. This is an important distinction because there is always a frame where the magnetic force goes to zero (the rest frame of the test charge), but there is not always a frame where the magnetic field goes to zero.

In fact, one of the invariants of the electromagnetic field is $E^2-B^2$ (in natural units). So if that invariant is negative then there is no frame where $B=0$. This is what is discussed in your second reference. But the point is that since there is not always a frame where $B=0$ you cannot consider the $B$ field to be merely the $E$ field as seen in a different frame. Instead both $E$ and $B$ vector fields are equally valid components of the overall electromagnetic field tensor $F$.

The electromagnetic field is governed by Maxwell's equations, which describes the origin of the $B$ field component as being due to current density $j$. This leads to your confusion:

But I don't know how you would explain the magnetic field due to its internal magnetic dipole through Special Relativity because the charge is not moving at all.

Actually, even a stationary electron has a probability current density. The probability current density essentially acts as a probabilistic velocity distribution, and hence the electron has a probabilistic electrical current density. The dipole moment can be thought of as arising from this $j$, with suitable QM mathematical machinery.