It may be worth enquiring about 'Training for Work', or at least that's what it used to be called. It is (was) a scheme by which if you were training to get yourself skills that would make you more employable you could get Jobseekers Allowance without having to sign on, for about 6 months if memory serves me right. The training provider has to tell the Job Centre that you stand a chance of employment at the end of it(with them I think), so you might have a problem at a big school, but if you are training at a small school and planning to get an FI rating and then work for them, you may stand a chance. Thus, you don't actually have to be available for work to get the payments. Ask about the scheme and then get an appointment with the resident expert in the Job Centre who knows exactly what you have to do to qualify and will probably look favourably on you if you are investing your own money to improve your chances of removing yourself from the unemployment figures.
Another point to note is that after you get your licence issued and 'officially' join the unemployment register you are only allowed to claim for about 6 weeks before you must broaden your search to include non-specific jobs (ie non-pilot)
Hope this helps
PP