There could be some other special velocity than c. Like, imagine there’s some special reference frame in which you can send superluminal signals at exactly 2.71828 c in any direction. In other reference frames, this special velocity depends on which direction you send the signal. Lorentz invariance is broken. But the only implication for local causality is that you need to make your bubble 2.71828 times bigger.
There could be some other special velocity than c. Like, imagine there’s some special reference frame in which you can send superluminal signals at exactly 2.71828 c in any direction. In other reference frames, this special velocity depends on which direction you send the signal. Lorentz invariance is broken. But the only implication for local causality is that you need to make your bubble 2.71828 times bigger.