BA have no reason to get bogged down in such silliness - leave the playground antics to the strikers.
Judging by how they have played this so far, I suspect they will play a waiting game and let BASSA \ striking cabin crew make as many of their own mistakes as possible. I certainly don't think BA will go for an injunction. It wouldn't benefit them to drag this out any longer. If strike action does take place I would suspect:
a) Sue Unite for losses on basis that strike is related to previous strike. This approach is a gamble with the courts but victory would mean it would be "easier" to sack whichever strikers they wished on the basis the Court would have ruled that the strike was "not valid". Allows sackings to be due to Union incompetence rather than BA bullying. As others have said, this will greatly sour the relationship with Unite but BA would then have room to make some compromises and build bridges directly with Unite if BASSA can be sidelined.
Or
b) See how how the strike goes. If it is poorly supported then just see it out, perhaps with targetted sackings of the "ring leaders". This is a gamble as sackings could be seen as bullying by BA rather than due to Union incompetence. Personally, I don't see BA taking this option as it potentially drags the dispute out for another 3 months and \ or opens them up to a heap of bad publicity.
It might well be Unite can somehow pull themselves back from abyss but with BASSA pushing everyone ever closer and TW having shown a complete lack of balls so far (will LM really be any better), lets hope that those cabin crew who voted for strike action are aware of the potential consequences.