Strix,
I am a musician and music teacher, with several descades' experience both of teaching in different circumstances and overseeing primary school peripatetic services. And I don't think I've ever read a question on a forum that has such a straightforward answer. You HAVE to keep her having the private lessons. If the kid argues, give her some chocolate or something and tell her "too bad". She'll thank you for it when she actually makes progress on the instrument.
Obviously a lot more progress can be made by an able student working 1:1 than in a group with others less able. This is exacerbated by the fact that on the violin specifically, learning good intonation (ie playing in tune) is excruciatingly hard, and requires constant, detailed attention from the very beginning to have any chance. That's going to be severely compromised by being in a group with others playing out of tune.
But we haven't even touched here on the main advantage of the private lessons over the groups at school, which is your ability to be involved in the lessons, liase with the teacher and support her practice appropriately during the week. This is probably the MOST important factor determining success vs drop-out rate with young children taking up an instrument - more important than so-called "talent", I would dare to say. It's one of the strengths of the Suzuki method for example, and it's one of the main reasons so many peripatetic school services fail.
Please don't be put off by the fact that you're a "musical novice". If you take a constructive attitude, listen to the teacher and are willing to learn, you'll support her well. In fact being a novice is even more reason why the school group would be a bad idea, because you'll have no way to judge the odd comments here and there being written in her practice diary (if you're lucky enough to even get that).
I DO know what your DC means about wanting to do it at school with her friends - playing with others is one of the great joys of learning an instrument. But it's a separate issue, which should involved joining ensembles, county groups, youth orchestras etc - or even playing the odd duet with a friend. The actual lesson in which the child learns how to play the instrument is separate from this.
The kid is able, keen, apparently on an instrument appropriate for her, and at the right age to be really going for it. You might be lucky and stumble across one of the minority of peripatetic services that serve musically able children well, but even if you were she wouldn't do as well as in a private lesson. More likely, she'll have some fun, waste a lot of time, and eventually give up because she can only play Twinkle Twinkle Little star by the time she's 11.
It's a no brainer. Find the money, and drop swimming or ballet or horseriding or macrame or karate or French or motor mechanics or whatever, as necessary.