Hi OP. Okay she's 6 and a violin player (this I know about, having one of those in the house). First of all, as I'm sure you know - violin is really hard. And it will take a while before it's fun. But it's a great instrument, trains the ear like nothing else, and a fantastic base for anything she may wish to take up musically in the future. And it's portable.
IME, if she's taking lessons at school, the expectations tend to be really low, so I would take with a pinch of salt the idea that "her teacher is very happy with her progress".
One thing really leapt out at me in your post. This whole "if she's not in the mood" thing. Really, I think you must knock this on the head. Who cares if she's in the mood? She must (and you must) get used to the idea of practise being routine, a given, a non-negiotable part of the day (just like eating, sleeping and brushing your teeth). Unfortunately what your DD has learned is that if she kicks up a fuss with screeching and tears, then she won't have to practise.
Haha's suggestion of having a clear goal is a good one. A warm up, a scale, and then perhaps focus on one part of the piece, for example. Please know that if you enforce a practise routine now, it gets easier (a lot easier) later.
I really want to encourage you and your DD OP. You've chosen the toughest instrument, but also one of the most satisfying. My daughter is pretty musical, but not exceptionally talented, and through disciplined practise has managed to become an accomplished player. I've also seen the positive knock-on effect of her music on other things. Stick with it - you won't regret it!