Gary,
There seems to be some system problems so I'll try this again.
All parameters of SR; time, length, mass and energy are routine in high energy researches. Momentum transfer is fundamental in particle accelerators. So if the conservation laws were in any way violated by SR, we probably would have heard by now. I know that's a non-answer, and your question(s) is a very interesting one. I think, however, that it goes more to the dual characteristics of wave and particle form associated with electromagnetic radiation. And that is of course subject to either a classical or Quantum approach.
EMR exhibits both a hyperbolic function and a parabolic function. The sinusoidal curve of the wave form is a hyperbolic function, and Minkowski identified the skew of SR transforms with hyperbolic function. Yet the absorption in the photoelectric effect results in a ballistic curve which is a parabolic function. And ain't that a fine can of worms!
So, yeah, what is the difference between refraction, reflection or absorption? And how can a massless particle or wave of energy carry momentum? The energy is there.
Personally, for myself, I have found myself in general agreement with Constantinos Ragazas, though it took me a while to understand so. Briefly, when the photon (sic) encounters matter, it slows from light velocity to relative rest at a time dependent exponential rate. That 'stacking up' of energy as it slows accumulates to become manifest as momentum. So conservation holds, the energy content of the photon equals its inertia.
Inclusive of that event are the peculiar properties of the near field and transition zone, the implication of how those measurable properties at even macroscopic distances might be definitive of field volumes, and how those might differentiate refraction, reflection and absorption. And this all flies in the face of QM and the photon being in superposition until observed.
I'm afraid that the best I can offer. For what its worth, jrc