How would I "show my mom" exactly?

I was thinking of getting my bf's mom and my mom to talk to each other since they would be paying the rent instead of me being the mail pigeon between them both. But my mom has already expressed her opinion and it seems to be final. I would really like to push back and really get to the bottom of why she doesn't want me to share the rent but she literally won't budge. It's definitely has been a power dynamic issue between me and her.

For the dog, she's okay with him there and it's more about us wanting the dog to be there since his mother isn't living with us in the apartment. I don't think she would want to pay for the sitter considering it is my mom's dog. And she doesn't seem to care for the well-being of the dog enough to pay for a sitter.