I am not a lawyer, this is just my layman's understanding from reading cases. Consult a lawyer for a more definitive answer.
You have to worry about defamation. For example, if you name the dog after a living female celebrity (substitute Jane Doe for this discussion), and then your characters joke about "Jane Doe" being a bitch, a slut, a dick licker, she likes to smell buttholes: Then you are defaming the real Jane Doe.
Even though you claim it is a work of fiction, even if you claim it was all in good fun and even the character's were joking.
You cannot "trick" your way through some technicality into being able to defame a living person by disguising the defamation as jokes or "about somebody else". If a jury concludes what you did would appear, to any reasonable person, to be intended to defame the real person, you have defamed them. Whether you think it was a joke or not. The same goes for disclaimers: Say all you want that nothing in your book is intended to defame anyone: It won't make a difference. There is not magic get-out-of-jail-free paragraph or claim you can make.
Maybe you can get away with stuff like that if you have lots of on-staff attorneys on staff, like certain TV broadcasters do. I wouldn't try it.
The only reason I can think you would want to do that is to make a joke about the real celebrity, or draw some comparison between them. Generally, since readers would understand that comparison, you are talking about the celebrity so they may have standing to sue you. Even "one name" celebrities that are widely recognized (Cher for example) might convince a jury you could only be talking about them, especially if a comparison was made. Finally, being complimentary in the comparison is not a defense.
If no comparisons or references are made, use a different name. Respect the celebrity's ownership of their name, brand, trademark, and public recognition.