Well, if you want advice, this is the policy for Somerset, which I can't imagine varies wildly from that elsewhere, given it's the law:
"Issues relating to shared residency arrangements
Difficulties in the application process can occur where shared residence arrangements are in place and parents/carers of the child submit 2 separate applications for different schools. In this situation the LA would ask parents try to resolve matters between themselves, and then inform the LA which application should be processed. Where possible the LA should not be involved in private disputes.
The LA does recognise however, that there may be situations where parents cannot reach an agreement between themselves and it is necessary for the LA to reach a decision. Where this is the case the LA will try to establish the childÂ?s permanent address, as set out above.
Each parent will be required to write to the LA and inform them of the number of days each week the child spends with them. Where the child spends equal time with both parents the LA may ask for additional information including who is in receipt of child benefit. Once the LA has received all the necessary information from both parents a decision will be reached based on the evidence provided."
Your complaint has no basis and may prejudice the appeal panel against you, I would imagine.