AITA for not letting my wife name our dog something stupid?

I'm new to Reddit, so hopefully I do this right.

I, 32 M, have been married to my wife Natalie, 30 F, for 4 years now. We have been living together with our 2 cats, Viktor and Warwick, since. The cats were mine before we got together, both of which I named after my favorite video game. Well, since Nat and I have been living together, she has been begging for a dog. Nat grew up with a Golden Retriever which she adored, and she had been pleading for us to get one. I was hesitant at first. I've never been much of a dog guy and I was unsure how the cats would react to having a dog around. Well, this Christmas, as a late Christmas gift, I caved, and surprised Nat with a Retriever puppy. I gave her permission to name the dog anything she wanted.

Bitch. She chose Bitch.

She said it'd be funny, and she'd always thought naming a female dog 'Bitch' was comical. I found it less funny. Not only would vets look down on us, but every Sunday, Nat and I babysit the neighbor's kids, 8 F and 6 M. What do we say when the kids ask about the fact the dog would have a collar with 'Bitch' on it? I thought it was stupid, and told her no. She told me I had just said she could name the dog, which is true, but I said I paid for the dog, and should at least approve the name. It's our dog after all. She said I was an asshole for taking away the freedom to name the puppy. AITA?

UPDATE: Thank you all for the kind words and advice. My wife ended up finding out about this post through a Youtube short, which I find funny, since I never thought this post would blow up. I am at work now and we are trying to work it out over text, but my wife is still being pretty stubborn and she doesn't like how people are talking bad about her on Reddit now. Hoping we can reach a happy medium.

UPDATE: I wanted to clear a few things up. First, my apologies about the spelling error. It is fixed. Second, when I say 'give permission' I should have worded it 'name the dog without my input.' Usually when naming a pet, it is in agreement of the two people. All I said to Nat was "What are you gonna name her?" And third, when I say 'begging for a dog,' I know she could get her own. I just don't like dogs and when she tried to convince me, I'd say no usually. Mutual agreement, again. I saw some people saying I was in the wrong because of the way I worded those parts of the story. So, I clarified a little. Anyways, we have since talked about the dog name and are still in a tiff. She is refusing to budge on her stance. I might just have to give in, as much as I don't like the idea