6.23 g. History of acute allergic reaction to fish, crustaceans, shellfish, peanuts, or tree nuts including the presence of a food-specific immunoglobulin E antibody if accompanied by a correlating clinical history.
It appears from your post that this is the disqualifying condition.
As I understand it, typically a challenge test (ingesting nuts) will be conducted to see if nuts can be tolerated. The results provide the SA with objective information. As a result, it should not matter a great deal who administers the test provided the medical provider is aware of what the requirements are in terms of passing the nut challenge test, or in other words, will administer the test to the waiver authority's standard. This is why it may be better to see a provider with some experience in providing the information needed to obtain the waiver as opposed to a privately obtained provider who may be less familiar with the SA's requirements in terms of information.