I know a bit about the legal framework, but have no experience of how the Adjudicator works in practice, so I hesitate to get drawn into the Kent discussion.
I will make just a few observations here.
Schools' Adjudicators were appointed under Section 25 of the School Standards and Framework Act 1998. Like Appeal Panels, they come under the supervision of the Tribunals Service.
Admission arrangements can be very contentious (!), and there has to be a way of resolving disputes which does not necessitate
recourse to judicial review every time. (Objections to the admission arrangements of Academies are dealt with by the Secretary of State, rather than the Schools' Adjudicator.)
Although the schools under discussion have foundation status, the Local Authority has every right to be be consulted about their admission arrangements and, where appropriate, to object. The Admissions Code states:
Admission authorities must consult with their local authority, all other admission authorities within the relevant area, the Admission Forum, admission authorities in neighbouring local authority areas, the relevant religious authority (in the case of the admission authorities of faith schools) and, for admission arrangements for entry in 2010-11 and subsequent years, relevant parents and other groups with an interest in the local area (for example, community groups).
If the LA has made an objection, they will have submitted the sort of form you can find here:http://www.schoolsadjudicator.gov.uk/object3.cfm
(click on "Objection form" under "Relevant documents").
It's important to distinguish between a 'campaign' and an 'objection'. If I were you, I think I would try asking the LA - or your local county councillor - for details of what the objection was based on.