If you haven’t already, send your MP your case history and any responses you have had. Include a covering letter outlining your complaint, and ask them to contact HMRC on your behalf to dispute that you should be made responsible for the error.
We have lots of evidence that active involvement of your MP goes a long way to a successful outcome, but actually the minimum you need them to do is use their letter-headed writing paper, as it does seem to make a difference to how cases are handled.
Your MP does have to act on your behalf; it is their duty. They have access to a specialist MP Tax Credit Hotline to help them deal with your case quicker, so tell them to use it. If your MP does not act on your behalf, advise them that you will make a complaint against them. It’s probably quicker and more convenient for you to deal with your MP (or their staff) via telephone or email. Both contact details should be freely publically available (to find out who your MP is go here: Find Your MP).
At this stage the MP will probably want to have one last try to reason with HMRC and will therefore write to HMRC on your behalf. This isn’t strictly necessary, but let them if they do it. Otherwise ask them to forward your dispute to The Adjudicator's Office, which is the next step:Dispute Step 4