Get over it. It is trivial in the grand scheme of things. It has been dealt with. You did not complain to the Head, it sounds as if you immediately made a formal complaint to the governors so that is why the Head will not have responded to you. It was about her so she let the govs deal with it.
The governors dealt with it and the school has changed how it operates.
Schools often receive letters of complaint. TBH yours is about an incident that wasn't well thought out but not serious. Governors are volunteers. Getting panels together is not easy on top of all of the meetings they attend and committees and working groups. That is why the complaint has to be very serious before the chair will amke that decision. They discussed it and decided the way forward. You would like a panel from another school to review it- I imagine that will not happen.
You wrote to the DFE???? FFS do you think the education system has nothing else to do.
You could ring OfSTED of course and report the school. You could say it was a safeguarding issue as children were upset, deceived, the school behaved immorally by lying to them and it caused long term harm and they would inform the local safeguarding officer who would invesigate. Tha sounds a worthwhile use of thousands of pounds of their time over an issue already resolved.
Do you just want an audience, a committee to sit infront of? It sounds like it.
Making a complaint does not mean you then get your own way. You are coming across as an obsessed, interfering person who can not keep this is proportion. It is not that important. Let it go or if you think the Head and the govs are not running the school appropriately, move your child somewhere else.
Schools do get things wrong. You would probably have been better off getting an appt in the Head's diary through the school admin person and having a chat with her.