OP I think you are doing the right thing, you seem to truly care about Betty's feelings and still want to be welcoming and polite. I hope Bill is reasonable and thoughtful in noticing that A) Carol isn't invited and B) it would be wholly inappropriate to bring his new girlfriend to his STBX-wife's best friend's child's party, especially if she hasn't specifically been invited. Its disrespectful to Betty, but it also puts Ann & Al in an awkward position in explaining to their own children who Carol is and why she's there.
As for the whole debate of social units, I come down very firmly on the side that carol most certainly is a mistress and most certainly is not a social unit with a married man. In my state a couple must be separated for 12 months before they can divorce too, and I don't think its unreasonable to say a person must wait that year+ to be in a new social unit - marriage after all is supposed to be lifelong commitment. To be able to divorce at all is a huge thing, to not have the maturity to wait it out at year or 3, as you break the lifetime legal and social (and possibly religious) commitment you made, to me is not something society should embrace. One should not rush into marriage, and once one is one, one needs to realize it is a serious and long term thing that can't just end in a moment's notice. If it takes 20 years to divorce, well so be it, be happy you don't have to wait for death and only have to wait those 20 years. And if you don't like that, don't make a lifelong commitment to begin with.