Hello PonderingFellow. Looks like you already got some great Qs and comments from the other posters

I am sorry you are going through this conflict. Something I'd like to point out is that (if I understood correctly) your gf has done nothing of note other than not perform cartwheels when your sister is around. And yet, your sister accused her of "trying to infiltrate" her life "in order to bring (her) down" and your mother accused her of "trying to divide the family." There's a fairly stark contrast there, right? Do you see it? Your gf is quiet or subdued around your sister but your sister is using military op language to describe her and your mother isn't too far behind. They sound paranoid honestly or hostile to outsiders? And your sister's friend (in reporting on "likes"...I mean, really??) sounds like she is

Have you ever introduced a different gf to them? How did they respond?

You said you agree with your gf that your sister is "very rude" etc yet you expect your gf to be pals with her? Why?

Some people are definitely more sophisticated in those awkward situations than others but it seems as though you're finding fault with the wrong party. As long as your gf shows a basic level of respect, I don't see an issue on her end. Are your family members showing her basic respect? Did they ever kindly welcome her into the fold? It doesn't sound like it.

I'm going to throw out an idea here for you PonderingFellow. I don't know how old you are but if you are old enough to seriously date and bring a partner home....it might be time to cut the apron strings! Do you know what I mean by that? You go out in the world, choose a partner, bring that person to meet your clan, and expect them to make her feel welcome...not judge her up and down and accuse her of military style ops. If they don't welcome her with open arms, you have two choices:

1. take a long step back from the clan and think about how you want your adult future to go...might it be time to set some boundaries with the clan? ("Mother, Sis, I love you and I also love my gf. You need to start being friendly and welcoming to her. When you respect her, you are also showing your respect for me. I know it's time to visit again when my relatives are being kind to my partner.")
or
2. dump the unwelcome gf in order to appease your mother and sis and prepare yourself for a lifetime of Mama Bear and Sis rejecting your partners and making all of your important life decisions for you.

I encourage you to consider Option 1 That and/or consult with a psychologist about your family and dating concerns.

Good luck