Who said anything about violating Lorentz invariance? The simple statement I gave not only entails Lorentz invariance, it also requires translation and time translation invariance, giving the actual symmetry group of SR - Poincare invariance. Tachyons do not violate this. They only violate causality, which is not inherent in either this formulation of SR or that in terms of Einsteins original postulates.
Again, tachyons do not violate Poincare invariance; so, you are simply wrong here.
I think you misunderstood my statement. The parameter c is a frame-invariant constant no matter how you formulate SR. My statement was simply that it only corresponds to the speed at which light actually travels because photons are massless. Were they not massless (and, strictly, we don't actually know that they are exactly massless - only that their mass is smaller than 10-18 eV) their speed would vary with energy; but, in no frame would it be equal to c.
As for relativistic mass, it is a deprecated concept in the practice of physics, as it adds nothing of any use, but tends to lead to a great deal of confusion. The problem is that relativistic mass is really nothing more than energy divided by c2. But, if we think about it that way, we can see that there is a perfectly consistent way to look at the kinematics of tachyonic particles if we just allow their masses to be imaginary valued. Then, since the Lorentz factor will also be imaginary valued, energy and momentum will be real valued and quite sensible.