I guess it depends on the history of the name. My mom rarely had the same last name as me and my sister and it was never an issue. I never really worried about it. Never felt left out, etc. My sister and I were happy to shed our surname when we married, so even had Mom gotten it changed, it would have been temporary.
Some women never take their husband's name. Friend of ours never took her DH's name, and their daughter has his last name, no hyphens. Since you didn't mention the circumstances with the first last name or how old the kids are, it's kind of hard to guess. Why are you having a hard time changing the name from theirs?
Personally, I'd take the fiance's last name and give a non-hyphenated name to the baby, but that's me. Me, who took back a birth surname as a second middle after I got married (ditched maiden, took DH's for general purposes).
However, please don't saddle the kids with a bunch of hyphens. They're still your kids, same name or not. I would also not change their last name if there are positive connections with the family it comes from. Let it be their choice.