Abraham-Lorentz force/electromagnetic mass/relativistic mass I am struggling to reconcile the concept of relativistic mass, to uncharged bodies.
So first of all I'd like to state that the concept of relativistic mass is derived under the assumption that maxwells equations are true in every inertial frame, and thus the correct frame transformation is the lorentz transformation. If a body is un charged, how do we know that the same rules apply? As for a neutral object, we cannot test whether or not maxwells equations apply.
Radiation reaction:
When charges are accelerated, energy is propagated from the charge at the speed of light. In order for energy conservation to hold, the concept of a self force was created.
See: https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force
The radiation reaction causes a recoil force coexisting with the applied force, with the relativistic generalisation being the Abraham-Lorentz-Dirac force.
Electromagnetic mass:
As a result of this effect, the concept of electromagnetic mass was introduced,
"Electromagnetic mass was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy is contributing to the mass of charged particles."
Aka, charged objects seem to have more inertial mass, than neutral objects.
This idea is confirmed by experiments done by Thomson
"It was recognized by J. J. Thomson in 1881[1] that a charged sphere moving in a space filled with a medium of a specific inductive capacity (the electromagnetic aether of James Clerk Maxwell), is harder to set in motion than an uncharged body"
Transverse mass:
Key equations put force by lorentz,
$ rest mass, M_{em} = \frac{2}{3}\frac{q^2}{ac^2}$
Where A is classic electron radius
$M_{T} = \frac{m_{em}}{\sqrt{1-\frac{v^2}{c^2}}}$
relativistic mass:
The effective electromagnetic mass looks awfully similar to the relativistic mass.
With the exception that in this formula, when the body is uncharged, mass does not increase to infinity as the speed of light.
Questions:
Relativistic mass is derived under the assumption the Lorentz transformation holds, experimentally for uncharged objects we don't know this is true (since it is derived from Maxwell's equations, if q is zero then we have no experiment).
Could this mass increase effect be purely a consequence of the interactions with the electromagnetic field? As lorentz formula when $q = 0$, seems to then ignore this effect,
If not. Which I suspect will the consensus,
Where do neutral objects come into play? From experiments, objects that are charged have more inertia than uncharged objects. But the standard relativistic mass formula seems to completely ignore this effect, as it is the same for charged and uncharged objects?
I feel that the resolution SHOULD be, that the charged objects have more rest mass than uncharged, in order to make charged objects have more inherit inertia, but this is not true. (unless we consider field energy?)
 A: First, the concept of relativistic mass has been discarded by the scientific community for decades now. Einstein wrote against using the concept back in 1948. Okun was particularly strong against the concept in the 1970’s, and particle physicists don’t use it and describe particles in terms of their invariant mass, particularly in their databases of particle properties (eg CODATA).
So any struggles you have with relativistic mass should be simply chalked up to one more reason to abandon the concept. It is pretty much only kept alive today by pop-sci authors wanting something to “wow” their audience with.

is derived under the assumption that maxwells equations are true in every inertial frame, and thus the correct frame transformation is the lorentz transformation

While it is true that this is one way to derive the Lorentz transform it is by no means the only way. My favorite alternative is to assume only the principle of relativity. From that alone you can derive that there are only two possibilities: the Lorentz transform and the Galilean transform. Then it is a simple matter of experiment to determine which is right.
Similarly, you can not make even the assumption of the principle of relativity and just write down the most general linear transform possible. Then you will get a bunch of free parameters that you can determine through experiment. It turns out that you can use this approach to deduce the Lorentz transform to within about 1% from just three experiments: Michelson Morley, Ives Stillwell, and Kennedy Thorndike. Of course, modern experiments make it much more accurate than 1%.
Furthermore, even Einstein’s derivation did not assume Maxwell’s equations were the same in all frames (that is a derived result). It assumed only that c is the same in all frames.

If a body is un charged, how do we know that the same rules apply?

We know this experimentally. The most convincing would be the data on relativistic neutrinos. For example, in the 1970’s there were several experiments at Fermilab that showed that neutrinos have a limiting speed of c, regardless of the energy of the neutrino to within 12 parts per million. This means that not only do neutral particles follow the relativistic transformations, they do so with the same invariant speed c. So the invariance of c is a property of spacetime that is reflected in Maxwell’s equations, not the other way around.
There are also tests using neutral kaons and neutrons. However, those may not be as convincing as the neutrino data since neutrons and neutral kaons are composite particles that are neutral overall but are composed of charged quarks. If those are considered then you could also consider most macroscopic bodies which are neutral overall but composed of charged particles.
A: 
If a body is un charged, how do we know that the same rules apply? As for a neutral object, we cannot test whether or not maxwells equations apply.

We can, and all experiments are consistent with Maxwell's equations. They don't stop working just because neutral bodies are around.

The radiation reaction causes a recoil force coexisting with the applied force, with the relativistic generalisation being the Abraham-Lorentz-Dirac force. Electromagnetic mass: As a result of this effect, the concept of electromagnetic mass was introduced

No, this is incorrect. Electromagnetic mass is a separate effect from the LAD force, it is not due to the LAD force. Both effects are due to interaction between different charged parts of the charged body: electromagnetic mass is due to that part of this self-interaction that is proportional to acceleration, and the LAD force is the next part that is proportional to derivative of acceleration. There are other parts proportional to higher derivatives of acceleration, but they are often ignored as these effects are extremely weak in practice.

Could this mass increase effect be purely a consequence of the interactions with the electromagnetic field?

No. The idea that inertial properties of charged bodies are purely due to EM interactions between different charge elements was studied in the beginning of 20th century but it was never completely successful. Some non-EM forces holding the charge together have to be present, and these forces are most probably associated with additional modification to energy and rest mass, so the resulting mass isn't purely electromagnetic.
For example, electron's mass was once considered purely electromagnetic, due to interaction of its hypothetical charged parts at distance called classical electron radius, which is around 1e-15 m. But from experiments on scattering and electron g factor we know electron is many times smaller than that, and possibly a point particle, so its mass cannot be purely electromagnetic.
The relativistic mass effect (or less provocatively nowadays, momentum being $\gamma(v) m v$ instead of $mv$) is universal for any massive body/massive particle, regardless of its charge state; and it is function of speed: the higher the speed, the higher the ratio momentum/speed $p/v$. At zero speed, this ratio is called rest mass of the body.
Electromagnetic mass is a different concept: it is the modification of the rest mass of system of charged particles (as manifested in changed acceleration under known external force) due to EM interactions between different charged parts of the body, when compared to a system where all particles have the same rest mass and configuration, but are uncharged. The body does not need to have net electric charge; a pair of positive and negative particles is a system that has (negative) electromagnetic mass. It is usually a very small effect; it is measurable e.g. in that hydrogen atom is somewhat lighter than sum of mass of proton and electron. The greater the interaction energy, the greater the mass defect.
A: 
Aka, charged objects seem to have more inertial mass, than neutral objects.


This idea is confirmed by experiments done by Thomson

What does "charged objects have more mass than neutral ones" even mean?
If we use a scale to make a 1 kg neutral object and a 1 kg charged object, then we have two objects with the same weight and the same inertia. Even if Mr Thomson would disagree. : )
A one ton charged elephant has more inertia that a one kg neutral cat. In this case I do agree that a charged object has more inertia than a neutral object.
If we were to neutralize a charged elephant by dumping some massless  oppositely charged particles on it, then the elephant would be exposed to some kind of radiation, the radiation being those charged zero-mass particles accelerated by the electric field.
Now this is something that Mr Thomson would not know, but an object gains mass, when exposed to radiation of massless particles. Except when the energy of massless particles comes from the object that the massless particles hit. Then mass of the object stays constant.
After the neutralized elephant has cooled down to its original temperature, then it has less mass than originally, yes. So that's the best example of a charged object having more mass than a neutral one, that I can create.
Now what was the problem again? Something about relativity. So we should now accelerate the elephant to high speed, in its charged state, and its neutral state, and compare the inertias.
Hmm well, the thing is that classical electro-magnetism does not have energy that has inertia, but it has this "charged objects have more inertia" - thing instead.
And for relativity it is the other way around. Or it should be.
If the thermal energy of the elephant has mass, as we saw earlier, then why would the kinetic energy of the elephant not have mass? And also the kinetic energy of the electrostatic energy of the charged elephant?
