A question: Is your sister also planning to invite people who are not family to this party? You don't mention that either way. Is it possible that sister's invitation list includes people such as elderly friends of your mother's, friends from church if that's mom's thing, friends from mom's neighborhood or wherever she lives--? If that is the case, then possibly sister is focused on a more formal event and/or she is worried about kids being too much for non-family guests, or she's embarrassed at the thought of a kids' skit in front of old family friends or the church pastor or whatever.
I am NOT saying sister is right here, though. She may be vastly overthinking this and be far too focused on what she sees as a formal party -- someone said, it's not a wedding, but is it possible your sister is seeing it as nearly as momentous and formal, since this is a big milestone birthday? You mention the kids being in the basement at other parties etc. but is this party at her home or somewhere like a nice restaurant--? It's possible sister is so invested in a perfect event for those who are not immediate family that she's forgotten to ask your mom what mom wants. Or if she did ask, your mom might have said "whatever you think is best, dear," as moms sometimes do.
In your shoes, I would talk to your sister ASAP and in person if you can -- phone only if you're not nearby. Text or e-mail is a no-go because she cannot hear "tone" in those. Tell her very kindly that you appreciate her willingness to do the party, but you're confused by the request and ask if she clearly asked your mom who mom wants on the guest list. I would really recommend that mom, sister and you put together the guest list as a team, so you can be sure mom's wishes (even if those wishes really do turn out to be "no kids") are being heard by sister. But only you know if inserting yourself into the planning is going to cause friction.
If sister balks, I'd tell her very nicely but clearly that your DDis not a "child" but a teenager who would love to help out at a grown-up party. That could be good-- offer to have DD help serve appetizers or drinks or whatever. I'd then agree that the skit is not happening but tell sister that your DD won't be comfortable telling her cousins that, and that sister should speak directly to those cousins' parents. Don't agree for yourself or your DD to be the go-between and the "heavy," as you put it. The reiterate that just because the kids are present, that does not make the party "about" the kids. I think she sees the skit as the part that will put all the focus on the kids, so do without the skit but tell her she has to be the one to speak with other parents about that.
Don't just take over the party. A post below suggested having at your house, your way, but if your sister is already committed to doing it, you will create even more bad blood if you announce you're doing THE birthday party.
It seems vital to get your mom involved and be sure mom understands that sister wants to exclude the kids. That is fine if it's sister's party and mom is on board--IF mom knows about it.