My girlfriend wants me to get rid of my dog because her cat doesn't like him

I (32M) have had my dog Rufus for 6 years. Hes a golden retriever, super friendly, well trained, basically my best friend. My girlfriend Amy (29F) moved in with me 3 months ago and brought her cat.

The cat hides from Rufus constantly. Rufus doesnt even try to chase or bother the cat, the cat just doesnt like dogs. Ive tried everything to help them coexist - separate feeding areas, baby gates, gradual introductions. The cat still hates Rufus.

Last night Amy said we need to "make a hard decision" about Rufus. I thought she meant rehoming the cat since she just moved in and Rufus has been here for years. Nope. She wants ME to rehome MY dog.

Her reasoning is that her cat is "traumatized" and cant live in fear. I said Rufus literally ignores the cat and the cat can access the whole house through cat doors and climbing structures. She said thats not enough.

I told her absolutely not, Rufus is family and was here first. She said she thought I loved her and would "prioritize our relationship." I said I do love her but im not getting rid of my dog who Ive had for 6 years because her cat is antisocial.

She gave me an ultimatum, either Rufus goes or she goes. I said I guess shes going then. Now shes crying saying she cant believe id choose a dog over her.

Am I the jerk here?? I feel bad but also Rufus is MY dog!