First: I'm assuming your GF is on medication to help deal with stuff? And is also in therapy? Perhaps this is something SHE should be trying to work out in therapy - not you having to "solve the problem". It's *your* life and *your* cat, not hers.

I've met phobic people who don't like animals before. They just don't come over to my place. So here are your options as I see it (in no particular order):

1) She can move out and you two can spend time outside of the apartment.
2) She can stop trying to manipulate you and realize that the cat is staying.
3. You wait until the cat's life has passed and just don't get another one.
4. You can dump the girlfriend.
5. You can rehome the cat.
6. You can train your gf to deal with the cat. Desensitization works wonders for phobias.
7. You can move into another place where you can more successfully separate your gf and your cats.
8. You can buy/create some sort of barrier to put the cat into where it's the "can't see them, can't hear them, thus they don't exist" phenomena
9. You can have a sit down chat with the gf and ask her specifically what she wants you two to do about the situation. Make it clear that getting rid of the cat is not an option.
10. You can find another gf who likes animals and thus it wont be an issue.

My suggestion?

This is something SHE needs to work out in therapy. It's your life and your cat. If she chooses to not go along with what you want in your life - then she'll leave. Because if you bend and compromise what you believe and what you want in this situation for what *she* wants... you'll wind up resentful and hurt. And if the situation stems from the fact that you no longer have the cat, and she leaves because you're resentful and hurt - then you're left alone. Which is not cool.