I think it's easy to see your twins or your triplets as part of a pair or part of a three.
Fwiw I think it's very healthy to see twins as two siblings who share a birthday, and triplets as three individual people who share a birthday.
When seen from outside the family, they are seen as individuals, and it's easy to think from inside that they constitute a job lot.
The muscateers all for one and one for all seems intuitive and reasonable from inside the family, but this is rarely how they are seen from the outside..
The thing is, if your dd who isn't invited is miffed, then by all means help her to overcome that feeling by listening to her.
The assumption that as she's a good pal of party girl so she's entitled to an invite seems strange to me. Party girl can invite whomever she likes, and she's expressed a preference (with a limited budget)
You need to let them all see the email, and understand that it's up to them what they do about it
For the girl who maybe won't enjoy the party, she might decline, for the girl who might enjoy the party she might accecpt, and for the girl who is close to the party girl and can neither decline or accecpt, she needs to learn that she's not entitled to an invite, just because she thinks she should get one- sometimes things don't work out for some reason.
Have a family discussion, round the table, and take notes.
Let them all speak in turn and try and be calm about it.... it's just a surprise, quickly organised party, and on a budget too.
If you are all declining, make sure to get a present for the birthday girl, and invite her over soon.
Party girl may assume that her relationship with her uninvited triplet pal is set in stone, and would be unharmed by excluding her and inviting her siblings, if this isn't the case, maybe uninvited triplet could send her a card saying "I'm sad I can't be there with you for your celebration, could we bake a cake together next week?".
The point is, is the friendship more important than the snub? And if it is, work on that and facilitate it.