I don't want my girlfriend to join my family group chat.
Why would you not want your girlfriend to join the family chat?
I feel she is invading my family and has no business there.
I can understand how you feel about this. Family conversations are expected to be only for family members. It might make things more complicated having your girlfriend join something so private.
I started the family group chat and I have the power to let her in.
Well it is good that you get to decide to let her in or not. Family conversations are expected to be only for family members. If you don't want to let her join that is her choice, but you should explain to her your reasoning.