Info here:
Making a Claim
Issuing a claim online here:
https://www.moneyclaim.gov.uk/csmco2/index.jsp
Court fee (to issue a claim) is dependent on the amount of the claim, for £3000-5000 it is about £100 I think. You could also have to pay a hearing fee if it actually goes to court. Other side pays your costs if you win (gotta love the English system

)
Downside is, if the court finds that it was your fault, then you'd be worse off than a 50:50, which is what it seems you'll get from your insurer. Also, if it couldn't be proved either way then the court would probably split it 50:50 and you'd pay your own costs, so as I say, only worth doing if you have a good case. If it really is your word against theirs, with no other evidence, then take what the insurance offers!