You do realise of course that the jury is still very much out on learning styles? Professor John Geke for example has produced an interesting study called 'Vak or Vakuous" in which he considers the fact that 'kinaesthic learning' is a load of old baloney. Indeed, my college has recently abandoned the whole VAK styles due to other studies which seriously undermine the whole theory. So I wouldn't base your argument on that at all. I don't think you can reliably justify your daughter's behaviour on her 'learning style' and think actually you are confusing the whole learning style theory full stop.
I also think you are confusing the two issues here (behaviour and problems learning to read), whilst almost certainly connected, you do need to look at both not just one, maybe I'm reading your post wrong, but you do seem to be dismissing the problem of her behaviour. Obviously, her behaviour will affect her progress, but in turn her behaviour may well be caused by her lack of progress.
I think if I were in your position I would approach the school and ask how you could work with the school to support your daughter in improving both her behaviour and her learning. I think this would be more likely to keep the teacher on side, and would be more profitable.
That said, I agree that I don't like the idea of having the mats just for selected children. I would say to the school that your daughter feels it is a naughty mat, and she feels as though she has been 'picked on' because of it. I would probably request that if they work, that all children are given mats (that is what they used to do at my daughter's preschool - you can get old samples from carpet shop for free).
Good luck with it, I hope you find a way forward for your daughter.