You're putting barriers in your way that may not exist.
A British solicitor (obviously one specialising in immigration matters) will know how to meet the requirements of the BRITISH ECO.
He will probably just draft a statement to be signed by the child's biological father and then returned. (Delivered and collected by courier if you want to speed things up).
I presume you have the father's address ?
Anyway, clearly you're in need of proper legal advice, but from a British immigration lawyer.