It is TW that has egg on his face at the moment. He publically stated that the last offer was the best that could be expected, his branches told him it was not good enough and sent him back to BA with his cap in his hands like Oliver asking "Please Sir, can I have more?"
BA does not need to do anything. TW is the one that is in a difficult position - will BA trust him with any further offers? The last one was conditional on several things, one being a recommendation from the union, TW gave that recommendation and his underlings made him look foolish and out of touch by refusing to recommend the offer. TW is the one that needs to make things happen to restore his authority and reputation.
BA on the other hand have the savings they need, they have not given a pay rise to the majority of crew, have introduced MF and can continue to grow MF as quick as they like without interference. They can transfer whatever work they like to MF without interference from the union as there is no agreement for route transfers for them to breach as UNITE have failed to negotiate one. There can be no court challenge to the route transfers as there is no agreement to breach.
BA can sit fat, dumb and happy in the knowledge that every day that passes moves a strike away from the holiday period and the savings they are reaping continue to grow. BA has no pressing need to settle this dispute immediately, they can afford to wait the unions out. If BA do act it may give the unions a new legitimate reason to strike, at the moment BA can sack people that strike as the action would not be protected.
AMICUS, the bit part players in the IA with only 40 strikers have effectively ruined the latest chance for a settlement - democracy in action? 20? Reps, looking after the interests of 40 CC, out of 13000 dictating whether there will be a consultative ballot on a settlement to the dispute. You couldn't write this stuff - nobody would believe it.