Your path would be the easiest to take, but somehow i doubt the Kenyan govt is going to assist in helping remove its favorite son from the Presidency. Of course there may be Empire records in London that can be accessed, but I think the actual document itself would be very useful to this line of questioning.
What I suggested is official protocol. We and Kenya are signatories to a reciprocal agreement, they must respond with documents to prove or disprove officially. That the U.K. Gov may have original documents is secondary, and we would not, I think, go there as a first resort, it’s not diplomatically appropriate.
The authenticity of the document would come into question only if proof was shown that Book and Page and Registration numbers did not belong to 0bama II.
Of course, they could dance around it, but, a Republican Senator could hold up any funding for their Country, too. As well, if they did, and it did prove to be true, they would risk our ire in future dealings with our new POTUS.