Clearly the assessor knows nothing about what a real headache can do.

Since you're in the UK I'd suggest you look up Multimedia File Viewing and Clickable Links are available for Registered Members only!!  You need to or which is a UK based CH organization complete with phone helpline. I expect they will have had lots of people with experience of going through something very similar, so they will be able to give targetted advice from what has worked / not worked for other people.

If you are in the UK, sorry, I have no idea.  In the US, there is no listing in the social security blue book for headache disorders.  You have to get a diagnosis for depression, anxiety, and suicidal tendencies in order to get disability.  A group of us were in Washington in April with an event called "Headache on the Hill" (HOH 2013).  We were asking for headache disorders to be included in the listing of disabilities.  Check it out on facebook.  You will find lots of information and help there.