Without giving away too much, category 4 is children of another faith living within the priority area.
Khushi12 - dipsydoo22 is right. Normally you don't have to complete an SIF for a faith school unless you are claiming priority on faith grounds. The SIF for this school is also primarily about faith. The few other pieces of information they request is stuff they should get from the LA. So it seems very strange to insist that non-faith families complete this form to get higher priority.
Given it is a secondary school, our understanding is that they do not need to have fixed numbers
I'm not sure what you mean by that. PAN (the total number of children being admitted) is a fixed number. The number of children in each category should also be a fixed number or there should be a clear formula for calculating the number. They cannot change the numbers on a whim. They have published numbers for each category in their admission arrangements. They have to stick to those numbers.
Their published admission arrangements state 152 for faith 2, 58 for faith 3 and 12 for faith 4. They used the same numbers last year and the year before. The explanation for the error really does not make sense. The numbers in each category should be the same this year as they were last year (and, indeed, the year before - and they are going to be the same next year) and are not dependent on the number of applications received. The explanation of the error does not make any sense at all.
I would go back to the LA and query their response. Point out that, according to the school's published admission arrangements, there is a fixed number in each category which is not dependent on the number of applications and the numbers were the same this year as for at least the last two years. Therefore, if the LA used last years numbers, the result should have been correct. Can they please provide more information on the error.
At the moment, based on what I can see, this is starting to feel like there was no error. The error has been correcting a non-existent error by withdrawing places from people who were correctly offered them. They cannot do that. However, it is still possible that there was an error and the problem is that the LA has not done a very good job of explaining it. But they still shouldn't have taken your place away.