Well, the overall point is:
If FTL communication is possible, then you get grandfather type paradoxes, unless you have an absolute time (or it is resolved some other way). And SR does not rule out absolute time, it just discards it.
If absolute time has physical effects, which it must if it can prevent FTL grandfather paradoxes, then yes, SR does rule it out. SR is Lorentz invariant, and absolute time preventing FTL grandfather paradoxes would break Lorentz invariance.
That doesn't mean you couldn't construct a theory that included absolute time and in which FTL travel was allowed but grandfather paradoxes were prevented. But such a theory would not be consistent with SR. To be viable, such a theory would have to match the predictions of SR in the domain in which those predictions have been verified, but it would necessarily make different predictions from SR about FTL travel. (No such theory exists that I'm aware of; I've seen hand-waving about it in discussions of LET, but never any concrete theory.)
My personal prediction would be that such a theory would end up being falsified (either that or rendered meaningless by experiments showing that no type of FTL travel is possible at all). But that's a separate question.
You're right, it would violate Lorentz invariance, but only for FTL travel. In that sense it is consistent with SR for speeds <= c.
I think it's quite easy to make such a theory, I'll get around to writing a blog post on it some day :)
The main result is that for communication (or travel) faster than c, invariance is broken, and it becomes possible to distinguish the absolute reference frame.
