Go for them in the Small Claims Court to recover all your money and get any additional payments you are due, including your time and costs in making the claim. Assemble your case, under UK and EU law, quote the relevant regulations, and file it. It just needs to be clear, NOT a "lawyer's submission". Attend the hearing, say your piece if asked, and you will be treated fairly under the law, not BA's idea of what they must do.
At no time go near a lawyer.
A judge will give short shrift to nonsense about weather and so forth, when the truth is plain.
It's inexpensive and works. If BA offers to settle beforehand accept only what you have asked for, and if they are not very, very polite indeed, as in "Please accept a voucher for a free Business Class trip for your and your partner as well as this cheque" add 20%.
Good Luck!
PS I know that the term "Small Claims Court" is not accurate, but everyone knows what is meant by it. Google for the details of what to do.