accord03, I do sympathise. Getting a job at the moment is by no means an easy thing.
However, I am a little puzzled by your first post. You say that you can't get a job until you start on T and you can't start on T until you have completed a name change. Am I right in this interpretation? If so, it costs only £33 to get your name changed by deed poll, and then you just take the certificate to your bank etc to get them to change your name on your official docs. I am not sure how it lies with the HMRC for tax purposes, but I am assuming that the procedure is the same. Show evidence of a name chnage and they will change your name. It is a hassle, but it is not a big investment if it is one of your bars to work and transition.
And I am assuming that you will get your T on prescription. So, whilst you aren't earning, that should be free.
Sorry if I have misunderstood, but I hope that helps.
Good luck!