Hello, they are not a company like most insurance companies so they would not be regulated by the FCA. They are a government body and a member of the Council of Bureaux.
The ways to make a complaint, is as you have stated, detailed on their website.
T: 01908 830001
Address: Motor Insurers’ Bureau, 6-12 Capital Drive, Linford Wood, Milton Keynes, MK14 6XT
As per their website, they provide a response within 10 working days.
If you are dissatisfied with this you can complain to the Chief Executive of the MIB, and if you are still dissatisfied you can write to the Minister for Transport at Department for Transport Licensing, Roadworthiness & Insurance Division, Great Minster House,***** London SW1P 4DR but you must follow the above complaints procedure first.
I hope this helps, and would be grateful if you could take a moment to provide a positive rating, kind regards, John.