My kids were 10 and 8 when I had my first diagnosis. I wore bandanas into school. They surprised me how resilient they were and I think they coped really well. However if your son is sensitive then I see your point. Kids, being kids their friends will probably know anyway and I found my kids got amazing support from the school and their peers.
Your reasons at work were exactly the same as mine. Not all my clients knew and it saved lots of explaining and let me just get on with my job. I felt I would never get any work done otherwise. I met a lady at the look good feel better course who told me that she was exhausted at the end of her work day as she spent so much if it explaining to others about her scarf and ultimately her cancer.
Wishing you lots of luck.
Paula x