The school is a small village school but they have 6 teaching staff and have not hit the class size rule with the total number of pupils on the roll.
If it's an infant school, each class is limited to 30 pupils, bar a few exceptions, and new pupils can't be admitted until the class size is below 30. If year 2 has 25 pupils but year 1 has 30, they can't take a new year 1 pupil, even though there are 5 spare spaces in year 2.
The places were turned down as full but not marked as infant class size refusal.
This is interesting. What does their admissions policy set the PAN (Published Admission Number) as? If it isn't an ICS case, then it goes on prejudice and that is much easier to appeal.
The best appeals are objectively strong and child-centred. Don't say things like 'we like the school' or 'it's convenient'. Look at what makes this school best for your child.
Find out, first of all, what the PAN is in the class you applied for, and whether it is 30 (or 15 if it is a mixed-year class setup).