A tricky problem.
My first piece of advice is no good to you, as that would be to not admit to migraine in the first place, particularly if you believe you have it under control. I had the 'third degree' from the RAF in the seventies when I admitted to pre-teenage headaches at interview. However, they let me in and I was free from migraine symptoms until my early forties.
As I now have a class one medical and migraine symptoms, I do have some information to share.
The first bit is not too helpful. My AME has shown me the CAA guidance, which suggests that, as you are finding, initial issue is problematic. However, if you are diagnosed after initial issue it is possible to keep the class one (I am, nevetheless, taking my own advice from para 1). So the fact that a class one can be kept should give you some cause for optimism.
If you go down the route of testing, the prices I was charged are as follows:
MRI scan - £560
comprehensive Neurological tests (hearing & balance) - £610
Consultation with top vestibular expert - £150
The CAA are much more pragmatic than they used to be, and I understand that their top doc is an ENT chap, so perseverence may pay off. Nevertheless, I'm still following para 1 and saying nothing. Good luck!