You write that you don´t want your therapist to stop acting naturally with her dog in front of you. When I read how she acts towards this dog during your session I personally felt it´s inappropriate for a therapist to share focus between a dog and a client.

Does she bring this dog as part of the therapy? I mean some use animals to address issues and to help the client to express him-/herself in an easier way. But if this dog doesn´t have anything to do with therapy I find it a bit strange that she brings it there.

I now understand that you can´t or don´t want to switch therapists but I would myself have a hard time accepting this. I would recommend you to try to say something of what you´ve wrote here, just a single line about how you feel without making a too big deal out of it. To me it feels that she spends way too much focus on this dog instead of you as you describe she calls it "dear" and "darling" when you sit there in front of her. You mean she talks to her dog while you speak?

I find this unprofessional but perhaps this is more common in other countries, in Sweden where I´m from, I´ve never read about a therapist who brings a dog to therapy unless the dog is part of the client´s treatment.