"A reliable source has told me that BA are recording the claims made by Unite and intend to use them in a future court case. Apparently, if they can prove that untrue statements affected the finances of the airline, then they have a valid claim against Unite.
Can anyone with legal knowledge say whether this is possible?"
Can't say, but I would guess that the same argument would also apply to:
A CC member who does not strike, but can prove detriment if they are eventually stuck with poorer terms than originally offered by BA, or
A striking CC member who argues that they only struck because of the misinformation put out by Unite, and suffer consequences of strike action