My daughter was born in the UK and I am British, but my wife had limited leave to remain in the UK.

I had difficulty in obtaining a Chinese visa to travel to China for my British-born daughter with an English name. She has a British birth certificate, but my wife didn't have permanent residence at the time. She got it 8 months later. Now my daughter is 10, the Chinese Consulate in London insists that my daughter has to obtain a Travel Document to China and my wife should invent a Chinese name for her. How will a Chinese official will allow my daughter to leave China at the end of the trip if there is no visa in my daughter's travel document? The names in her Travel Document and in the British Passport will be different as well.