How old is she?
Who looks after her mostly ?
I wouldn't let her take the dog. She will just never return her.
However much our dogs mean to us they are property like anything else.
To recover this dog she will have to make an application to the county court for a declaration of ownership and a consequent possession order which she will not get because you have clear rights to this dog.
The risk though is that a Judge would say she is a joint asset and order her sale and the profits divided between you which nobody wants.
If I were you I would just make sure she cannot lay hands upon this dog. If she does the onus will be upon you to recover which is hard to do.
If you keep the dog under your control then it is very unlikely that she will sue.
Can I clarify anything for you?
You could do that.
I think it is clear this dog is jointly owned so it wouldn't matter much. There would be an advantage is setting down the purchase details in writing.
If she doesn't deny it then that is as good as an admission.