If you go for a private allergy diagnosis, often the NHS will not "believe" the results.
Also it's possible the private diagnosis will not be in your NHS medical records if there was an incident of allergic reaction and a hospital doctor had to look up your child's notes.
I know you are very upset about what has happened but it sounds as though you really need to take all of the emotion out of the situation and only look at the facts.
From your posts, I think the facts are the NHS have diagnosed your eldest child with several allergies and have prescribed anti allergy medicine for him and have given you a care plan.
Have a copy of that with you when you see the GP.
Shortly afterwards, a NHS professional made a mistake and told other agencies you did not have medical confirmation of the allergies, Do get a copy of this in writing so you can also give it to the GP.
If that's the case, then it should all be cleared up.
Then go back to your GP, see whichever one in the practise is most likely to listen or change GPs if you feel no-one in your current healthcare practice will be impartial enough to give you a fair hearing.
Ask for an urgent allergy test appointment for your youngest child.
In the meantime, could you arrange for someone else to be there - family, friend, neighbour - whenever you feed your youngest child a potential allergen, so that they can be an independent witness to any reaction and could accompany you to A+E.