Since the child is well into grasping her musical foundation, isn't she old enough to choose for herself which she prefers? A lesson/talk with a violin teacher and one with a fiddler...followed by a little bit of time to think it out on her own ought to solve the 'problem' for all of you--girl gets whichever she feels drawn to and the adults don't have to feel snippy about it because the bottom line is that you both honestly want the child to love music.

It seems to me that fiddlers don't care whether you call their fiddle a Violin...violinists curl up their nose if you call theirs a Fiddle.

She's learning to read/play by reading music for the piano.
Fiddle should encourage her to broaden into also playing by 'ear' Violin keeps her papertrained.