Sorry, perhaps I'm not explaining it well enough. If your claim is that BA is lying about weather, which I think is your submission, and all the evidence points towards weather disruption then you will lose because that's extraordinary circumstances.

Had you claimed that BA cancelling was indeed due to the weather, but they failed to take all reasonable measures to get you to destination by alternative means such as other airlines, that would likely have been a much more difficult thing for them to defend.