@olly91 Some of this is normal, expected "rudeness" from that age group. I'd largely just let it alone. Things like "my house is bigger and better," you can respond with "yes, houses come in all shapes and sizes because everyone has their own preferences. My husband and I bought this house because we really like [insert thing you guys like about your house]. What is your favorite thing about your house?" If she says "our car is newer" or "we spent $ on dinner." You can just respond and acknowledge what she said. Don't really put any emotion into it. "Yep, that's right your car is newer." The point is to model that these objective "facts" are just that, facts. They have no bearing as to how you live your life or the pride you feel in your own vehicle and such.
Now of course there is time to intervene - you're not going to let her open your daughter's birthday presents. You're going to say "its [your daughter's] birthday so she gets to open her presents. Remember on your birthday YOU get to open your presents." If she pushes back you will probably just have to say something like "our rule is that the birthday girl gets to open her own presents. please sit with the other kids now."
But again, for other things, just let it go. If she complains about paper plates - "we're eating dinner on paper plates tonight - here you go." or that your daughter's birthday is at the park - "yep, daughter likes to have her birthday at the park." If she then says she doesn't like that? "Hmm well then you can have your birthday party somewhere that isn't the park!"
And then in the meantime, just continue modeling what you view is appropriate speech. And you can also (when this friend isn't around) have conversations with your own daughter about it. You can say things like "Do you remember earlier when [friend] said that her house is bigger and better?" and then ask questions like "Why do you think she said that?" and "How did that make you feel?" You can ask if your daughter thinks those were kind things to say?
I agree with your choice that since the friend wasn't listening to your nanny, she doesn't get to come over and play while the nanny is there. I'd also remain pretty strict with that for yourself - if she's acting up, send her home. "That's not allowed in our home. I'm going to send you home now, you can try again tomorrow/next week."
Right now, as frustrating as this all is to deal with as the parent, its pretty mild stuff. You're generally able to supervise and you can intervene when necessary as well as have conversations with your own child about this. This will be especially important if you see this friend acting in an unkind manner to your daughter. You can talk about healthy friendships and all that. I think this will also be a good lesson in that different families/people value different things. And that's OK, as long as everyone is generally being kind and respectful. Basically, I'm saying that this child is offering you a good opportunity to have these conversations and build up skills that your daughter will need as she grows up, especially because you'll likely never be able to influence your daughter as much as you can now. Eventually, your daughter is going to be interacting with peers who may display far more undesirable behavior, but its not going to be occurring in your presence. And you'll want your daughter to have some skills to determine if this behavior is OK, as well as the trust to come to you if she's unsure/over her head.