If your claims management company (CMC) demands you give them a portion of your payments when you succeeded claiming from the bank your refunds in full, then you have the right to complain or take legal action against them.
Claims management companies work on a no win no fee basis. If they succeed in making your claims successful, then a portion of your refunds goes to them. The CMC should leave you alone if they failed to make your claim successful.
CMCs such as PPI Claim and Advice will respect the results of your complaint whether we had succeeded or failed to recoup your refunds. If you had succeeded on your own after our failure, then you deserve to receive the amounts due to you in full.
If your CMC continues to harass you with calls, you can take up them up on a complaint with the Financial Ombudsman Service or the Financial Conduct Authority.
These companies can only work on your behalf with your approval. To avoid any issue with a CMC in the future should you make your claim successful, sign a contract stating that their service has been terminated. It should also state that any action you take that results in a successful refund is yours to receive in full.