Now, let's just start with what I do know and what I don't know:
I know that $$KE=m_0c^2(γ-1)=Δmc^2,$$ which is not too hard to derive using the Lorentz Transformations if you know basic calculus (correct me if, in the derivations, I have made some assumptions I shouldn't have). From this follows, then, assuming a stationary body possesses some energy $E_0$ and, consequently, that $$E=E_0+KE,$$ that $$ΔE=ΔKE=Δmc^2.$$
I do NOT know that the rest mass energy is, in fact, $E_0=m_0c^2$.
Assume that I also know everything that came before Einstein - i.e., Maxwell's equations, the De Broglie principle etc - and that I am not a very advanced physicist, to say the least.
Using this information, can you derive that $E=γm_0c^2$?
I know an article that is kind of trying to do just that, but I get a feeling it is deliberately over-complicating things to make everything as valid as possible; here is the article. That said, maybe, you could simply go through the same method as in the article but in a layman's terms?
Either way, any valid derivation which wholly proves the mass-energy equivalence identity for ALL scenarios without going any deeper than special relativity and empirical or intuitively explicable principles would do the trick all day long.
EDIT: this provides a very, very basic but, otherwise, a valid derivation for $E=m_0c^2$; however, in that derivation, one question arises: how can mass just disappear? I do realise that the mass must have been emitted as light in order to conserve the total momentum of the box, but how does that work in terms of changes in the electric field? You see, say, the electrons in the object emitting the light vibrate, thereby creating an 'ever-changing' (always non-zero second derivative) change in the electric field, which then propagates by subsequent alternating changes in the electric and electromagnetic fields, this propagation capable of 'the amount of energy spent on vibration of the electron'-worth of work - so where does the mass-to-energy transformation step in???
