Once you provided the insurer with the information required by the policy, which appears to be the name of the instructor and shop, it should be their problem, not yours.
I don't know how these things work in the UK, but there does appear to be formal process you can follow for insurance disputes over amounts less than £150,000 which won't require you to pay out for a lawyer.
If you're unhappy about the way your claim has been handled or the way your insurer has treated you, you have the right to complain.
www.abi.org.uk