Why leave it until there is one blank page?
A "fast-track 3rd party agent", if such a service even exists, is not going to be any faster than doing it yourself
I have decided that if I am not living in the UK 12 months before I expect my passport to become full / expire, the only way I am going to renew it is in person using the same day service in London
The backlog was quite horrible this time last year but I think the passport office has managed to become quite efficient now. Of course unless you have 2 passports / other travel docs you'd still be stuck in one country for 3 weeks or so.
The fast track system is only available if you are in the UK. I have seen various agencies claiming that they can handle a fast track application on your behalf, thereby avoiding the necessity for you to go back to the UK, but I have no idea how trustworthy or reliable any of them are.