By completely different reasons than the one described at the OP, which was to show the wrongdoings in Einstein's short paper, I have
one idea based on the original framework, but using only two photons emitted in opposite directions by a magical reason, with frequency f.
As it's a "thought experiment", I can redesign the original scenario by using only two photons which, curiously, were emitted by two different
atoms exactly at the same time and frequency, and exactly in opposite directions along the x axis (to cancel momentum).
Then, at the lowest energy level possible (as per Planck), the final equation for dE (Delta Energy) would have been:
Delta E = 2.hf.(Y-1)
and, as v << c, then (Y-1) = 1/2 v2/c2. Then, the total energy change would be:
Delta E = 1/2 (2hf/c2).v2 , which has the form of KE = 1/2 mx.v2
what would be the meaning of such a strange mass mx = 2hf/c2 ?
It's not mass lacking from the rest mass m0, because no mass was lost when both photons were emitted.
Could it be some kind of electromagnetic mass for the two photons, having each one an EM mass mp = hf/c2?
Because it contains the relationship E = hf = mpc2, given as an approximation for v << c.
What is v? And what is Y?