I owned several dogs with my now ex-partner. I purchased a dog from his bitches litter just before our breakup.  I paid for her, she is microchipped in my name but he has refused to do the change of ownership for KC. Is there anything I can do? How can I change her into my name? I don’t have her KC number, I only know her Kc name as I chose it. Any help would be appreciated

This sounds like a civil matter and something if he won't do amicably is to go through a solicitor. Whether as part of the split or the doglaw solicitor.
As being in a relationship you wouldn't have had a contract either I'm guessing?

Unless you want to show/breed your bitch, you really don't NEED to change the ownership with the KC.  Much as I imagine you'd prefer to have her registration papers in your name.  Might it help to talk to the KC if you know the name?   I have a feeling they'd want to see the signed transfer paper.