This mirrors my recent experience, when BA Customer Service claimed that my flight was cancelled due to "weather, air traffic control restrictions or other reason outside our control" whereas it was, in fact, due to crew being out of hours.

Here's what I did:

1. Replied to BA, asking them to reconsider their initial response. They stuck to their guns.
2. Followed it up with a letter (sent recorded delivery) requesting payment in 14 days or I would "take the matter further". BA replied to say that they will defend any legal action robustly.
3. After 14 days, initiated a MCOL claim. BA replied to say that they intended to defend the claim.
4. At the very last moment before judgement would have been made automatically against BA, they backed down and paid the claim in full.

So, don't back down. Just pursue the claim and BA will, eventually, pay out.

And... yes, BA has sunk this far. It's sad.