UPDATE
Hi again to all who kindly shared their knowledge and experience, and gave support and understanding. I wanted to give you all an update on the outcome, and also possibly to help other parents in a similar situation now or in the future.
I was very close to giving up, but as a last ditch attempt I contacted our MP, acting on advice from the thread. This led to our application being referred on to a specialist team within the Passport Office and based in Glasgow. They deal with cases where there are special circumstances relating to children. They contacted me, and after very thoroughly reviewing our application the passport was issued in my DD's name.
This meant that my DD was able to go on her school trip, and alongside her friends and peers she experienced something that will hopefully have a lasting positive impact on her life and education, (as well as having had quite a lot of good old fashioned fun).
I was treated with absolute professionalism, sensitivity, common sense and kindness by our MP's team and by the staff at the specialist passport team. I can not praise both highly enough. There was a recognition that not everything is black and white, and that our laws and rules need to be administered with that in mind, and with humanity and compassion. Who knew that such a specialist team existed. There is a strong case for all similar applications to be referred to them in future.
It wasn't just about getting the outcome we hoped for, it was about helping us on our journey to leave the past behind and recognise us as the family we are now and have been for many years. In essence, to shake off a shackle from the past, to feel more free and less trapped (hence the user name).
I would advise any parent in a very similar situation, (or a solicitor advising them), not to give up. Ask for your DC's application to be sent on to the specialist team who have the expertise to understand the complexities of such applications, rather than the generic passport staff who often do not. If the passport office won't refer you on, then contact your MP and request their help. You also don't have to wait to contact your MP until after you've exhausted the passport office complaints procedure. Before taking these courses of action however, please read the full thread. I was only able to get this help because of my wholly positive motives and our very specific circumstances.
My sincere thanks once again to those who helped 