I get the enforcement bit, but one pupil I saw elsewhere on the net had a pair of Clarks school shoes in leather in black, which only resembled trainers in some aspects of the shape, so looked fine to me. Also, I would carry my coat in my bag in some of the hot weather we have been having.
Years ago my DD was pulled up for wearing neat black jersey trousers which looked smart because they had a metal stud on the pockets, so the school said were jeans. There are a lot of grey areas, unfortunately. School attendance is compulsory, so I personally think the school needs to be outside of the grey areas before they should be allowed to deny a child a day's education. I am not a stroppy parent either, more an ex teacher,l ol.