How can she explain while using proper etiquette?

My parents have been raising a puppy for two years, that was bought by my brother's ex girlfriend. The reason they have it to begin with is bc when they broke up, she couldn't have a dog at her new place. My brother has moved on with his life and with a new girlfriend, but the ex and my mother especially have always been on good terms and cordial to each other. My mother however wants to break ties with the ex bc she is feeling like she's disrespecting my brother and his new love for having the ex over at times and to see the dog she left with mom. My mother told me the ex came over the other day and the dog didn't have anything to do with her. She sat on my mothers lap and loved on her and didn't pay no mind to the ex. Surely one would think the dog is clearly attached now to my mother after 2 years. My mother also has a few dogs of her own that the dog has got attached to. So in short, my mother has been the care-taker and mother to this little dog and has got attached herself. How does she tell the ex she wants the dog without making bad waves? My brother says he'll pay for the dog for my mom, but my mother says that she has paid for it 3x over already with buying it's food, giving her the love and walks etc...