The four year rule is clearly stated in the admissions criteria. It has been made publicly available. It is your responsibility to check it. You did not.
You were asked to provide the SIF which would have highlighted this. You did not.
Then asking the admissions board to take into account information at a later date when they had given you the opportunity to do so earlier, and even reminded you to do so is what goes against you I’m afraid. The Bucks website also tells you to read the admissions policy for each school you are applying to. They cannot make people read stuff - you had plenty of reminders to make sure you had.
I really do feel for you as it sounds like if everything is above board and if you had provided the information when asked then your son would have a place.
Your anger should, however, be with yourself and not admissions, the head or anyone else. We all make mistakes and unfortunately this one has had consequences that affect your son.
You need to look at an admissions panel likely response to “I made a mistake, I didn’t read the admissions policy, I didn’t fill in the SIF when asked, I provided the wrong date, but now I want to give you all the information I should have done when you asked for it originally and expect a school place for my son.” When everybody else given a place in similar circumstances obviously did manage to do this. Why should you get special treatment and different rules to everyone else?
I honestly don’t want to seem confrontational - we all want our first choice schools for our children. It’s an incredibly emotional time. Nobody would blame you for trying at appeal. Just be realistic that it doesn’t look good.