If the specific IgE to bees or whatever else is negativ, than there is no type I allergy, thus no risk of anaphylaxis.
If specific IgE is positiv, than there is in fact a risk of anaphylaxis. In this situation piloting an aircraft will probably not be possible. This is identical to the current situation with carrying epinephrine. So documenting a negativ specific IgE could well be the only chance into a flying future.
The concepts in immunology have advanced considerably in the last years
, and I doubt, a GP is capable of advising correctly in this case. At least where I live I would prefer somebody with adequate training and my experience from the NHS when I worked there suggests it is not better in your country
.