I claim PIP for a long term chronic condition that impacts my life hugely and I’ve needed carers in the past to help me when I’ve been mostly bed bound. It’s not migraine so I can’t help with specifics but can give more general advice. I have had chronic migraines alongside my other condition, so I do understand the impact.
In theory you absolutely should be eligible as it obviously impacts you more than 50% of the time. The assessors are supposed to adhere to the criteria that if you cannot do something reliably, repeatedly, safely and in a timely manner then you can’t do it. however in practice, it will be a battle simply because you have some days when you don’t need the support. Its unfair as it obviously impacts your life hugely but the PIP process is stressful and frankly unfair but it sounds like you already know that.
My advice would be to absolutely apply but do so with the acceptance and expectation that they will decline your first application but be prepared to fight it. The whole re application process recently was about 18 months for me so also be aware that it will take a long time.
Use that time to gather as much evidence as you can. I can’t stress enough how much the evidence is important. Yes they assess and ask how the condition affects you but unless they have medical evidence, they can and do ignore it.
Ask your specialist for copies of any recent letters, print out of your record from your GP, prescription listing medication, letters from anyone who provides care and support for you saying how they assist you. A diary for a couple of weeks to show the frequency of attacks anf the support you needed at this time.
All the information you wrote above about therapies you have tried is something you can add on the extra information section, as it helps build a picture. If any of the therapists you’ve used can write a letter detailing your difficulties then that may help.
Also absolutely get someone who is used to filling in the forms to help you apply. I used a local charity ( for my condition) and also an online benefits group. I know others who have used CAB or local disability groups.
I hope some of this helps as I know how incredibly hard it is to live with a chronic condition and to be so affected 15-25 days a month is without doubt a chronic condition.