I suffered from various allergies as a teenager - nothing ever identified as the cause, and something which thankfully I grew out of.
A side effect of having very irritated nasal linings as the time was frequent nosebleeds.
I had cauterisation done over twenty years ago as part of a nose and sinus operation.
It did the trick - never had a problem since.
I would not rule out that procedure - I was lucky to have had it carried out under general anathestic. But I believe cauteristaion on its own is relatively pain free.
I suggest you push your GP to refer you to a decent ENT Consultant.
Your symptoms suggest that it's fairly out of the range of what a GP can do for you.