As others have said here, inconsistent use of your middle name shouldnāt really be effecting your credit score, however I have went through the same proccess of removing mine from all things official.
The easiest way to do this is simply remove your middle name via deed poll, this is free and is a document you create yourself (witnessed by two people) saying you give up your previous name, you can generate this document here: https://freedeedpoll.org.uk/
Many companies will require proof of your name change, and this can act as it. Without this document I have had inconsistent results.