The name on the chip may or may not be the owner. In the same way that a motorcar V5 is not proof of ownership, the chip details are just contact details if the animal is found.
You are correct, the cat is not hers, the cat is yours because you never transferred ownership of the cat to her.
However if you go to the police, because she takes the cat, the police are not likely to get involved because of the issue with the chip details.
If she takes the cat, you can apply to court for an injunction (and as the court for court and legal costs) to make her return the cat to you.
If she simply takes the cat, by all means report it to the police as theft even though there are unlikely to do anything because of this chip issue.
Ultimately, if she takes the cat when you are not there, you are faced with making a court application.
My suggestion is to tell her that if she insists on taking the cat, you will make an emergency application to court to prevent her pending a hearing over ownership of the cat. You have plenty of witnesses who will attest to the Being yours and all she will have a few vets bills which he paid.
On the balance of probabilities, the judge would find the pet belongs to you based upon the facts you have given.
Can I clarify anything else for you?
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We can still exchange emails. Best wishes. FES.