She sounds really needy and insecure. That's not good in a person her age. Think about it: Does she really even have other friends besides you? If so does she treat them this same way? I would bet she doesn't have any close friends because she drives them away with this childish behavior and then thinks it all goes away if she says, "Just kidding."
I would sit down and ask myself what I originally liked about her enough to consider her a friend, and whether those qualities outweigh her neediness and insecurity.
Do you share interests? Is she funny? Smart? Do you have good conversations (that are not about her or her feelings that her sisters aren't paying attention, etc.) -- can you talk about the world outside her little domain and yours too? Does she do activities that you like as well?
Or is your socializiing limited, in reality, to just chatting about her and her issues; or other friends' issues; or just to partying that is about having drinks and not really sharing any interests or activites other than meals and drinks?
Because a lasting friendship needs shared values and interests, not just eating out and talking about....her.
If you and she really do share things that make her a friend and not a party/lunch acquaintance, then tell her next time she does this "you didn't invite me" routine: "You know, I really do like to spend time with you but I've noticed that often if I bring up something I've done with family or other friends, I feel like I'm just sharing my life with you by telling you about it, but you say what you just now said -- Why didn't you invite me? When you say that, I feel awkward and hurt. I like to see you but don't invite anyone but family to family events, and I'm sure you see other friends without me, and I'm happy to hear about that. I hope you can understand that I like our time together, but am puzzled about why you bring up feeling like I should have invited you. What's wrong?"
That final question puts the ball entirely in her court and she likely will splutter and say "Just kidding" or be embarrassed. (She should be.) But you need to nip this constant "pay attention to me" attitude. If you really do like her, tell her so clearly and tell her that you feel you can't SHARE things with her -- like what you did over the weekend -- because she will respond that you should have invited her too. Definitely wait for her to give you the opening with one of her "Why didn't you..." complaints, but be prepared mentally with your speech to use right then and there. If you can get her to see that she does this and it bothers you -- but you still like her -- it could change your friendship for the better.
If you evaluate your short friendship and find that she is not all that funny or smart or good to talk with, and you can't really share with her, and the relationship is based on going out in groups and just hanging, eating or drinking -- I'd keep distancing myself from her just as you're doing, because you're not friends, you're acquaintances, one of whom is too focused on being the center of attention.