I think as far as inviting herself to your other events, like the zoo, just keep doing what you're doing and telling her "no" or "not this time," and I think that's the moment to bring up inviting yourself if there is a time to do it. Lea might just have to learn this the hard way. I agree with words like "I wish you wouldn't have asked because now I feel bad telling you no."
I think one big issue here, which I probably don't even need to say, is that you need to have a discussion with your mother and your sister in advance, that Lea is not invited to this one. You do not plan on including extras in the BBQ on Labor Day, which means Lea is not invited, and please do not invite her, which includes not discussing it around her. If she is an "adopted daughter," it's natural to assume she's included, so you need to stop that before it starts on the occasions you do not wish to include Lea in the family gatherings. This is more important when you find that not only do you have an extra, but now you're left with a slumber party on top of it, that you're not particularly interested in doing, at least not this time. I think that this "teaching moment" lies more on the shoulders of Sister, as the "adopted mom," or your mom as "adopted mom" (not clear who did the adopting) or Lea's own parents.
In the meantime, I'm not sure how comfortable I would be having a discussion about inviting herself. You sound close enough you could reasonably do it, but I think I would be more inclined to just deflect and continue saying, "Next time," or "Not this time," etc., and have a talk with Sis about them inviting people without clearing it with you. I would hardly expect my sister to leave her kids or stepkids at home, and if Lea is a member of the family, it should be expected she's a part of the package. My kids, nor my neices and nephews, expect to be included in all activities other people are doing, friends/family, something Lea seems to not have picked up on, so something needs to be said. The big question is if it's appropriate for you to be the one.