OP, I'm a teacher.
Under under normal circumstances, I'd have a bit of an eye roll at days off for birthdays etc. But, given the circumstances you describe, I think it would be a lovely day.
It'll be unauthorised but, if his attendance is otherwise good, it doesn't mean anything in the grand scheme of things.
I've permitted my exh to take our children out of school for holidays when they were younger - only after the attendance data has been submitted to the LEA of course 😉
Not all education happens in the classroom.
I've spoken to the teachers on every occasion they must be sick to the back teeth of me, but only when I said I was going to move him to a new school did they refer him for the sessions
This isn't because they don't care (hopefully!) but because finding is limited and we have to justify every child who is referred. They will select the children who they feel are most in need.
If you have spoken to them many times and are dissatisfied, write to the Governing Body.
Like I say, I'm a teacher. I was a Parent Governor at my child's school and I had to write to the Governing Body about bullying that wasn't addressed. That was awkward but it had to be done.
Unfortunately, bullying can be a very difficult pattern to break.