For now - the message ought to be:
I'm doing this for you, because it's important to me that you feel relaxed and happy in our home.
It is possible you will get this in return: If you really loved me, you would choose me over the dog.
One response to that is: I get that. And I also realize that the dog and I developed some bad habits - including bad dinner table habits - when it was just the two of us. I understand why his presence in the kitchen and bedroom was stressful for you. Thank you for being patient with me while we worked out a compromise.
As for her using the dog in unrelated conversations, the best move is to see that for what it is. A weak persons power play. Expect it, and don't react to it. Stay on topic and stay calm. I'd make a large bet that if she runs that play 3 times in a row without any reaction from you at all, she will stop doing it without you having to call 'foul'.
That play, (dragging the dog into unrelated disagreements) only has as much power as you give it. The more agitated you get, in any discussion where the dog comes up, the more the subtext reads that: you love the dog almost as much or as much as your wife. The more relaxed you stay - the less worried your wife will feel about her standing in the house.
The core theme here is easy to describe, harder to execute: You are without doubt my highest priority, AND also not my only priority.
The more she feels, believes and accepts that theme, the better off you will both be.
turnera: The marriage is actually very good...which is what frustrates my wife and I. She admits that she is unreasonable with this dog issue and made a mistake by not realizing sooner, but she is very caring and considerate in most other matters. If the marriage wasn't so good, I would have picked the dog and dumped the wife months ago. I have spoken to some of my long-time friends about this issue and they feel the same way as most of you do about the dog. They think what she is doing is unfair and I should not let the dog go. However, they also warn me that I will have a tough time finding someone else who complements me as well as my wife if I give up this marriage. I could very well get a divorce, have my dog for another 8-10 years, and meet someone better. Or I might end up with someone who likes my dog but doesn't treat me as well. Or, I could give up my dog, be sad/upset about it for the next 10 years, but have a loving wife for the rest of my life. I guess I'm trying to be careful to not just assume that the grass is greener on the other side. No matter what I choose to do, there is a huge chance that things could end up worse for me.