Unfortunately, you can't control other people. It's kind to take into consideration that kids who are not invited may have hurt feelings when something is talked about in front of them (so try not to do that, obviously), but you can't prevent her from ever even knowing it happened, most likely. And, you shouldn't have to. Sorry, but that's life. Everybody doesn't get invited to everything.
It's not a fun thing to learn, but it's reality. If you want to invite the girl, fine, but without the younger sister. MOM can make the call if she wants to decline the invite, then she can explain it to her daughter. It isn't your fault. It's her mom. I say invite the girl, but make it clear to her mom that ONLY the one child is invited. If she chooses to decline (for whatever reason--sister isn't invited, kid has other plans, she's mad, kid has chicken pox--whatever her reason), then that is HER decision and not yours. No reason to feel guilty at all.
For what it's worth, I can't stand when a parent tries to tie one kid's activities to their other kids. Kids are individuals. Even twins, for goodness sakes.
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ETA, I also would start the conversation with the mom acknowledging that you are aware she said her daughter couldn't come if her sister wasn't also invited. Then tell her point blank that you *do* want the friend to come, but her younger sib is not invited. If mom wants to decline the invite, that's fine, but you didn't want her to feel excluded BY YOUR DAUGHTER. (it isn't your daughter that is excluding her, it's her own mom) This mom is trying to control other people so that SHE doesn't have to be "the bad guy" with her kids. If nobody invites either of them to anything, she doesn't have to explain to either of them why only one was invited. It's dumb, really, b/c kids understand this stuff really well, better than parents at times it seems, once you explain it. So she has made YOU be the bad guy (and your daughter by default) to exclude her child. That's an awful way to parent, in my opinion. And I wouldn't let it dictate to me what kind of person/friend I (or my child) would be. Invite the older daughter. She's your daughter's friend. And your daughter wants her there. Right? Don't invite the younger sib. Let mom sort that out. It's our job.l
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If your daughter chose 3 OTHER girls (that doesn't include this neighbor child at all), then that's fine. You behave politely, and try not to be obvious in talking about it in front of people that aren't invited. That's just good manners. Whether the kid lives next door or not. Sometimes it cannot be helped. But if it can, then do.