It's really common to have a tightness in your throat when you're anxious that feels like a lump. It's also really common to be unable to eat - that's part of the fight or flight response as our bodies don't want to be loaded up with food if we might need to run for our lives.
Your little boy is clearly suffering from very significant anxiety triggered by being in school. It sounds like he could be struggling with any number of things including the busy sensory environment, social difficulties, changes to routine and concentration. These need to be properly identified and acknowledged by school and adjustments made to help him feel safer and under less pressure.
Have you ever considered whether he is on the autism spectrum? If this is a possibility, it would be a good idea to ask the GP to refer him for a neurodevelopmental assessment. Autism could explain why he is finding things so difficult in school and feeling so stressed that he can't eat or sleep.
It isn't OK for professionals to say that a child is experiencing such significant issues as a result of anxiety without trying to find out the reason they are anxious and making changes to reduce them.
My DD2 has Asperger's and, when she was 7, (pre-diagnosis) I asked her to design her perfect, magic wand school. She designed a school with one teacher, no other pupils, animals to cuddle, her own bedroom where she could go for a rest, no food and no smells. That gave us lots of clues to what she was struggling with and some adjustments were made to help her on the back of it.
I hope that's helpful. Happy to share more info by PM if you would like.