There are some ultimatums that deserve considerations and others that only leave you considering "Should I actually say 'There's the door' or just point to it? " Being told "it's me or your pet" elicits the latter response.
This exact scenario happened to a user on Imgur. He reports that his former girlfriend reached out to him and offered to get back together... if he got rid of his cat, Hobbes.
One look at this cat, and the answer is pretty clear. NOPE. No deal. Though, not necessary in this case as Hobbes' dad defended his decision saying:
"I'm 33 years-old, and Hobbes is the closest thing I have to a kid."
Right on! Who would ask someone to give up their kid?
If we may, we'd like to offer a little advice to the girlfriend: If your not a pet person, cool. Do you. But know that about yourself before you decide to date a pet person. It's a bad idea to think you can change anyone, especially when it comes to pets. And certainly, don't ask the person you're dating to get rid of this pet. Chiefly, because that's a hurtful request for the pet owner, but also because the answer will almost always hurt you too. The pet will most likely win.