Chris
Worth a top score if just for the title! Very good and nicely written analysis. I believe I have valuable contributions, see my essay, and took up your challenge ref GPS a while ago; http://vixra.org/abs/1001.0010 (probably now in need of updating).
One shock was a revelation about the 'overwhelming evidence' for clocks falling out of syc, and the cause of apparent lateral Doppler shift, which is now both proved and DISproved! IEEE Trans. On Inst&Measurement. Vol 52. No.5 2003 http://ivanik3.narod.ru/Eather/ejo7t3n8Thim.pdf
I've published in the UK Skeptics magazine (2011) that Hafele & Keating were co-erced into omitting their actual atomic clock results, which were not consistent with SR, to get published. This is also mentioned, with Hafele's own quote, in Kingsley's essay here.
I agree with you that Einstein's solution to the effects of lateral shift which ARE found is illogical. There is a better solution, which recognises the quantum process of scattering at 90 degrees; The light signal measured is NOT the original signal. It is a new signal scattered (re-emitted by Raman/Compton scattering) from the particles CHARGED BY the primary signal. This has even greater implications. A receiver in relative motion cannot then use 'Proper Time' to estimate the original speed of a 'light pulse' from the sequence of individual scatterings from the particles. He can then quite validly find APPARENT c+v for that original signal relative to his own frame.
This is indeed hard to first get your head around, then suddenly everything falls into place, like solving a hierarchical 3D moving jigsaw puzzle. Surprisingly, the postulates emerge unscathed, but can now have a logical explanation. space-time and SR emerge direct from QM, if both slightly re-interpreted. It's all about the effects of the 'detection' process.
I hope you'll read my essay (slowly!!) and comment on the quite beautiful logical solution that emerges. You've made far better inroads through the quagmire than most so should grasp it, though it's slippery as an eel until the old assumptions are abandoned!.
Very best wishes.
Peter