OP, a/k/a Ann, here. A few points of clarification:
1) How do Ann and Al know that Bill and Carol seem to be fairly serious? In a word (OK, 2 words), social media. By this, I don't merely mean Bill has updated his Facebook status to "in a relationship with Carol," but regularly posts updates, pictures, etc. about the things they do together.
2) As there's been some question about how much contact the children have with Bill, Betty, and Carol, Al and Ann's kids (including the daughter whose milestone event is being celebrated) have never met Carol. Bill and Betty's kids have spent LOTS of time with Carol and her son, and seem to like them. (I personally think Bill and Carol should have waited a bit and dated each other one on one without the kids for a while first, rather than immediately rushing to make so many dates/ activities "family" outings with all the kids, but that's neither here nor there.) Likewise, Betty and Carol have seen each other (e.g., at B & B's kids' sporting events) and exchanged hellos, but haven't really spoken much beyond that.
As to why I haven't just asked Betty what she'd prefer, I already know. She'd prefer not to see Bill and Carol together at this particular event, but she's also been trying very hard to be a mature adult about the whole thing, and would never ask me not to invite Carol, because she wouldn't want to ask me to choose sides or potentially cause tension for Bill and their kids.
Sooo ... what I'm leaning towards after reading all your feedback (thanks!) is inviting Bill and Betty separately, along with their kids, but not inviting Carol. If Bill asks whether he can bring her along, I'll tell him Al and I would love to spend some time with them and get to know Carol, but that a good-sized party with our extended families and his S2BX probably isn't the best venue for that, and we'll understand if he doesn't feel like he can attend without her. Obviously, if he doesn't ask and just shows up with Carol, I'll welcome them both and extend the best hospitality I can (even if I have to complain a blue streak to Al about it once the party is over).