BA are insured by a range of insurers around the world. Are we talking about their EL policy in Australia, their motor policy in Russia, a policy for one of their planes, D&O policy...
In any event - if you've got a claim against BA, then it's BA you need to approach. Whether BA involve their insurers is a matter for them - and even if they do you may not actually know about it.
Whilst not knowing about the circumstances, the Small Claims Court is generally a pretty good way for the 'little man' to go about pursuing the 'big company'. It's cheap for you to pursue the claim, but expensive for BA to defend - hence they may be minded to settle on an 'economic' basis.
Why not tell us the background! We're full of advice and you'll get a whole range of comments!