Do you mean there are 31 in the current Reception? If so, I'm afraid that doesn't help you. It almost certainly means that one child was admitted as the result of an appeal and is therefore an "excepted child" who doesn't count towards the infant class size limit until September.
The fact that the school is under capacity also won't help you. If this was an ordinary appeal it would be useful but it is not relevant in an infant class size case.
The school has an admission number of 30. This will therefore be an infant class size case. That means the only way you can win is to show that the LA has made a mistake.
The childcare issues and the fact your daughter is shy won't help, I'm afraid. Even if it wasn't an infant class size appeal they wouldn't make a strong case.
The fact that the school may increase its admission number to 60 is interesting but probably depends on more classrooms being built to increase the school's capacity. Again, I don't think this will help you.
It may not say anything in the admissions booklet about checks but that doesn't mean there aren't any. You may want to write to the admissions team asking them what checks are carried out to detect fraudulent applications. They may, of course, not want to answer in detail.
The confusion over the waiting list is interesting. They really should be able to tell you how many children have been admitted from the waiting list and why your daughter wasn't admitted when there was a vacancy. This raises the possibility that a mistake was made and your daughter should have been admitted, but it falls short of the proof you would need for an appeal. Write to the admissions team asking for confirmation of how many children have been admitted since 19th April and why your daughter was not admitted in each case.