Doubts regarding Einstein's 1905 derivation of mass-energy equivalence This is a follow up on this question. 
In his [paper][2] under the title:Does the inertia of a body depend upon its energy-content, Einstein drives the famous $E=mc^2$ equation. 
His argument can be summed up as follows:
Consider an object(like lantern,so it's not a point particle) that is at rest in a given inertial frame. It has energy that is given by $E_0$. Let it emit two beams of light carrying the same amount of energy in two opposite directions, So that the body will stay at rest(since the light will carry equal momenta in opposite directions). Let it's energy after emission be called $E_1$. Consider this same exact situation from an inertial frame that is in relative motion with respect the the frame at rest. And denote the energies of the body before and after the emission by $H_0$ and $H_1$. 
He then considers the quantity: $H_0 − E_0 − (H_1 − E_1)$:
$H-E$ according to Einstein, is simply the difference in the Kinetic energy of the body up to an arbitrary additive constant $C$. So that:
$H_0 − E_0=K_0+C$,
$H_1 − E_1=K_1+C$,
$K$ is the kinetic energy of the body in the moving system.
He then states since $C$ does not change during the emission therefore:  $H_0 − E_0 − (H_1 − E_1)=K_0 -K_1$. In other words, the change in the energy of the body after the emission of light is given by the difference between the kinetic energy of the body before and after the emission. Now since the body does not change it's velocity, therefore He infers its mass has changed. 
There's one part about this argument that us unsettling for me. Namely the constancy of $C$ during the emission. 
If we have a macroscopic object, it's energy is given by(at least) the sum of its kinetic, internal(thermal) and potential energy.
That is: $H_0 − E_0=K_0+\Delta E_{T_0}+\Delta U_0$,
         $H_1 − E_1=K_1+\Delta E_{T_1}+\Delta U_1$.
In other words the constant $C$ denotes $\Delta E_{T}+\Delta U$. Assuming the $C$ is the same before and after the emission is tantamount to assuming that both internal and potential energy of the emitting body is constant during the emission process. It's not self-evident to me at all why this should be the case. Why say, the internal energy(temperature) of the body won't change after emitting light? Einstein did not justify this assumption at all in his 1905 paper.
So Is it a justified assumption? If so, then what is the justification?
 A: https://en.wikipedia.org/wiki/Einstein%27s_constant
Start with the last part:
"About constants
The Einstein field equation has zero divergence. The zero divergence of the stress–energy tensor is the geometrical expression of the conservation law. So it appears constants in the Einstein equation cannot vary, otherwise this postulate would be violated.
However since Einstein's constant had been evaluated by a calculation based on a time-independent metric, this by no mean requires that G and c must be unvarying constants themselves, but that the only absolute constant is their ratio:
G \over c^2 "
I don't pretend to be an expert.
A: 
This is tantamount to saying that the potential and internal energy of the object in a given frame is the same before and after the emission. How is this assumption justified?

Because potential energy is internal energy. It isn't some other energy in some mysterious other place, it's right there in the body. To understand this you need to start with Compton scattering, where some of the photon's E=hc/λ wave energy is converted into electron kinetic energy:
Image courtesy of Rod Nave's hyperphysics 
The point to note is that if you performed another Compton scatter on the selfsame photon, and another and another, then in the end, you take away all of the photon energy and there's no wave left. The photon has then been entirely converted into electron kinetic energy. You will hopefully agree that this is reasonable because "light is just kinetic energy".  
But note that we could have put that photon though pair production instead of Compton scattering. We could have used that photon to make an electron and a positron. So you could assert that the electron is quite literally made from kinetic energy. Remember electron spin and the wave nature of matter, wherein in atomic orbitals electrons "exist as standing waves". You could claim that in some respect the electron's mass-energy is "hidden" kinetic energy, and that the mass of a body is a measure of its energy-content. Think of photon momentum as a measure of resistance-to-change-in-motion for a wave moving linearly at c. What would you call resistance-to-change-in-motion for a wave moving  at c in a Dirac's belt spin ½ path? You would call it mass. You would say "the difference K0 − K1, like the kinetic energy of the electron (§ 10), depends on the velocity".  You would say the electron was a ody, and then you would look at electron-positron annihilation and say "if a body gives off the energy L in the form of radiation, its mass diminishes by L/c²." 
Note too that when you drop an electron, you would say that gravitational potential energy is being converted into kinetic energy. But all that's really happening is that internal kinetic energy is being converted into external kinetic energy, and as a result, the electron falls down. Then when you dissipate that kinetic energy as radiation, you're left with a mass deficit, because there's less kinetic energy in there. 
IMHO it's all very clear if you pay attention to things like Goudsmit and the discovery of the electron spin, and the Einstein–de Haas effect which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". Of course since Einstein didn't know that in 1905, you might claim that the assumption is an unjustified assumption. But we can trace the connection between light and matter a long way back, at least as far as Newton: "Are not gross bodies and light convertible into one another?" That they are. You can work that out by just sitting in front of a fire. And three hundred years later, you might understand why.  
