SR doesn't depend on behaviour of gauge fields. Special relativity is necessary to have a meaningful definition of "particle" in field theory. The gauge fields have to have zero mass term because of gauge invariance, not Lorentz covariance. The mass is generated by interaction with Higgs particle, this is essentially a trick which lets you forget gauge invariance after the model is postulated. It doesn't impose any requirements on SR either.
I was thinking of how Lorentz invariance was historically arrived at: From Maxwell's equations. If the photon has mass, then presumably Maxwell does not exactly describe its behaviour (although with the current upper bound it will be a very good approximation); but the underlying massless gauge field may still follow Maxwell.
http://www.nature.com/news/2011/110922/full/news.2011.554.html
http://arxiv.org/abs/1109.4897v1
http://usersguidetotheuniverse.com/?p=2169
http://news.ycombinator.com/item?id=3027056
Perhaps the end of the era of the light cone and beginning of the era of the neutrino cone? I'd be curious to see your probability estimates for whether this theory pans out. Or other crackpot hypotheses to explain the results.