Tom,
"Then what the heck is c'?" The same as c, but speed of propagation in a different place, in an inertial system. Exactly what SR says, but plus Local Reality.
Think of Doppler shift as it really is; a shift of the real scalar quality wavelength, only later computed against time to give the metaphysical 'frequency'.
As with sound waves this may then also be due to the change of speed required at the transformation, so this is the alternative we posit; Mutually exclusive real material inertial systems, then conforming with all exclusion principles. An 'inertial frame' is then simply a 'state of motion' of some particle or collection of matter. An electron, a cloud of electrons/protons, a lens, or any 'dielectric medium'.
The simple 'discrete field' Laws are; 1. 'All space is field' (again as AE said). It's just that some media (i.e. the ISM) are more diffuse, so have greater extinction distance. And; 2. All particles re-emit at c, whatever the relative 'approach' speed was.
Ergo; All observers always find light speed = c. And light speed always 'propagates at c' (SR) because, as in Copenhagen, the observer makes it so on detection (QM).
At the high ion shock densities propagated at very high speeds the Doppler wavelength change (blue shift) approaches gamma and 'Optical Breakdown' energy density (OB mode ~ 10^21 ions/cm^-3) so we get a 'hockey stick' curve, excess heat, the ions in the LHC hit the 'wall' (and astronauts loose radio contact as ambient ion density increases promoting the OB bow shock).
So there your are. No messing with the 'complete' mathematics, but a non-linear quantum version that removes all apparent paradox and logically predicts all the 'anomalous' effects found. Now I promise you I really didn't intent to be quite so 'ambitious' Tom, but it just is what it is. Once the kinetic code was broken it just all poured out, washed away the rubbish and clarified things!
It makes real the Minkowski/Einstein (1980/1952; "not one 'space' but infinitely many 'spaces' in relative motion." Only really the 'semi hidden' assumption in SR interpretation that 'no absolute background' also means 'no moving background' is falsified (an 1893 error). I'd characterise it as equal slight reconfigurations of SR and QM to make them one. Quantized GR by the way is implicit.
It is an intellectual challenge, but I have every confidence you'll rise to it Tom. Perhaps think of yourself as a litmus test for mankind!
Peter