Hi, I was in the same situation a couple of years ago; put in 6 schools thinking at least one school will offer my child a place and trusting local council to do the right thing, but no offers in all 6 choices and the council shoved my child into a school that was 8 miles away and notorious for social problems. I turned down the offer from that bad school, I was told that was the worst thing to do and putting my child at risk of having no school places.
The following October, my child was offered places in two of the schools who turned us down.
I can feel your pain at not knowing what to do and getting no help from the council or from the current primary school. I was so angry and sad, my child was OK because his mates were supporting him.
The end of my story was; I looked around the other neighboring boroughs and applied at their schools on waiting list and my child was offered a place in May by a mediocre school which was quite near to were we live. My child is now very happy at this school even though none of his mates from his primary school go there, bonding with the new found friends.
I did do appeals for all 6 schools who turned us down and I went to their panel meeting, all of them, also the council messed up my waiting list applications to the schools in the other boroughs.
In October, two of the schools that was on my CAF list wrote to offer my child a place since they have new vacancies and my child was on their waiting list; I turned them down because my child is happy where he currently at.
I really hope things will turn out better for you and your child, hang on in there and please don't give up, just do it and keep trying.