James,
I agree, viewed from that approach the equation wouldn't balance. It needs to be thought of differently, and the equations are shown to balance. We should probably start with the electron e, a high energy quanta with spin energy double any simple orbital angular momentum, so a 2-sphere envelope comprising a twin vortex torus would, for instance, work well. (Earth's magnetosphere and galaxy AGN's have similar forms). In non elastic scattering the e dominates the incoming photon amplitude (In fact we need to stop thinking of the photon as a 'particle', and just imagine the QM version, as a wave train or amplitude fluctuation distribution).
Now, the e experiences nothing of the outside world except what 'arrives' and then 'leaves'. The waves of a relatively weedy little photon are dominated by it, so whatever 'rate' they approach at (subject to relative propagation speeds) the speed IN the e is the same, c wrt e, so the wavelength lambda is not only dictated by the e but is the only one it 'knows!
Now to the re-emission. This is exactly the same process. The e does not know or care what happens after it re-emits each 'wave peak' at the only speed it knows (c wrt e) it doesn't care what happens to it, it just emits the next, and the next, at the same speed it 'absorbed' them at, which is still c wrt e.
Now the fact that that particular c 'after leaving' may NOT be the same as the c before arrival is no business of the electron (I call it the 'datum' for speed). which has no data on it. So, as far as the electron is concerned the accounts all balance perfectly. The emitted 'wave packet' will be more 'quantized' in amplitude, but just like the wave from a pebble in a pond, will spread again in time (Shrodinger sphere surface reduces by the ^3). That also wasn't all in the essay due to 'space' (lol).
Now step back yet again for overview and and consider; Nature does not wast effort. The whole point of the systems and the electrons job is to modulate c. All very simple. But one more touch of beauty in tidying loose ends;
What is true is that 'outside' the e lambda has changed, but of course frequency has also changed (in the propagation frame). The sums are then dead simple, and represent the Galilean Transformation, with 'gamma' growing towards optical breakdown frequency (NOT 'infinity'!) or minimum lambda (L). We then have;
Observed from at rest in the approaching frame; c = fL, electron frame; c' = f'L', then in the receding frame c" = f"L". That gives conservation of energy everywhere, the point of the whole business. Also remember, electrons are doing this job with all signals from all angles all at the same time!
Now if you are the observer and don't like undergoing all those accelerations to check actual propagation speed you can use apparent speeds from trigonometry and a calculator. Pick any state of motion. You'll find frequency appears to stay the same whatever the relative motion of the electron, so wave speeds 'appear' to vary. All you need do is add or subtract your own speed relative to the medium of propagation in each case and you will find the same REAL results as above.
Now the 'EM cross section' of plasma is virtually zero (it's 'dark') so the only giveaway is the kinetic effect. See the Emsellem et al formula in my essay. That's how we work out galaxy rotation speeds. Lateral motion and harmonic resonance must also rotat the wave charge density axis, so give tiny rotations of optical axis. This would provide the same effect we find and have many names for; i.e. 'stellar aberration, 'curved space time', 'refraction', 'diffraction', etc, all subject to local particle density, so very gradual in diffuse media such as space, but pretty instant in dense dielectrics such as glass (prisms). Faraday rotation of polarity, also not previously explained, is an accompanying consequential effect. (That's all just one bit near the middle of the jigsaw puzzle).
I know that's a lot to absorb (I must stop the puns!) but it does all matter!
Peter