Concerning their magical girl style, Miyu is all about logic and control, Ilya is all about images and fantasy, so it's quite understandable Miyu would time after time find it hard to accept Ilya as her fighting partner. In accordance with what Ryll said, I think Miyu also wanted to remove Ilya from the fighting for Ilya's own sake (fearing Ilya would be a danger to her own life as well, not just the others'). It's quite a wornout cliche, but they are little girls, so it would be fitting. Not to mention Miyu is socially awkward even during her best moments.