I don't know about this specific instrument (violin/fiddle), but in general I have observed that some individuals who learn music (instrumental or even vocal) under a rigid system remain stiff and unimaginative in their playing/singing, but many others do not.

I believe that some of us are better able than others to make music freely and instinctively, and that those who "have it" are not likely to be spoiled by formal instruction ~ on the contrary, a solid inborn talent is generally augmented, not negated, by good formal music education. At least, there seem to be many performers who combine the best of both worlds, so to speak, and I assume that such people had a natural instinctive feeling for music, and then when they had lessons, they were able to improve even more.

I am certainly familiar with the type of musician or singer who has a great deal of training but shows little or no "feeling" in their performance. I figure that such people would simply have nothing to offer musically were it not for their formal education ~ I don't believe that they had some spark of creativity that their musical education extinguished.

Of course, the younger the student, the greater your fear of "miseducation" ruining something ~ that's understandable. Since your daughter is already studying two musical disciplines (piano and dance) without harm to her natural rhythm and feeling, it may be safe to assume that adding lessons on that other stringed instrument won't hurt her.

Are you familar with the Suzuki method? I don't know a whole lot about it, but it seems to be an approach to the violin that works very well (especially with very young children) and develops the student's "natural" musicality first, before getting into the "nuts and bolts" of reading the dots, etc., and moving on from "Twinkle Twinkle Little Star" to serious classical compositions. If you find a good Suzuki teacher, that might satisfy Mom's desire for "violin" formality without endangering the young student's personal musicality.

If you have friends (playing partners) who play the fiddle according to the tradition you know best, be sure that your daughter at least gets to listen in, and has the chance to try her hand at session-participation, if and when she wants to.