The key to understanding what to do is in her answer to you: “I am not married or engaged to you, I don’t have to be close to them.” Let’s take a look at what this means.
If she is living with you and says the reasons she won’t be with your family are that she isn’t married or engaged to you, does that mean that she will be fine with them when she is? Unlikely. It seems she is using it as a way to validate her reluctance.
The other possibility is that she is completely telling you the truth about how she feels. Since she is not engaged or married she feels no obligation. Since her feeling about your family is clear, she is telling you there are very clear limits on the depth she is willing to be involved with you.
I recommend couples counseling for the two of you to get this issue on the table. If she is unwilling to do this, or can’t evolve from her current position, it may be time to rethink the relationship.